Linear Regression and Correlation

## Testing the Significance of the Correlation Coefficient

OpenStaxCollege

[latexpage]

The correlation coefficient, r , tells us about the strength and direction of the linear relationship between x and y . However, the reliability of the linear model also depends on how many observed data points are in the sample. We need to look at both the value of the correlation coefficient r and the sample size n , together.

We perform a hypothesis test of the “significance of the correlation coefficient” to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population.

The sample data are used to compute r , the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But because we have only have sample data, we cannot calculate the population correlation coefficient. The sample correlation coefficient, r , is our estimate of the unknown population correlation coefficient.

• The symbol for the population correlation coefficient is ρ , the Greek letter “rho.”
• ρ = population correlation coefficient (unknown)
• r = sample correlation coefficient (known; calculated from sample data)

The hypothesis test lets us decide whether the value of the population correlation coefficient ρ is “close to zero” or “significantly different from zero”. We decide this based on the sample correlation coefficient r and the sample size n .

If the test concludes that the correlation coefficient is significantly different from zero, we say that the correlation coefficient is “significant.”

If the test concludes that the correlation coefficient is not significantly different from zero (it is close to zero), we say that correlation coefficient is “not significant”.

• If r is significant and the scatter plot shows a linear trend, the line can be used to predict the value of y for values of x that are within the domain of observed x values.
• If r is not significant OR if the scatter plot does not show a linear trend, the line should not be used for prediction.
• If r is significant and if the scatter plot shows a linear trend, the line may NOT be appropriate or reliable for prediction OUTSIDE the domain of observed x values in the data.

## PERFORMING THE HYPOTHESIS TEST

• Null Hypothesis: H 0 : ρ = 0
• Alternate Hypothesis: H a : ρ ≠ 0

WHAT THE HYPOTHESES MEAN IN WORDS:

• Null Hypothesis H 0 : The population correlation coefficient IS NOT significantly different from zero. There IS NOT a significant linear relationship(correlation) between x and y in the population.
• Alternate Hypothesis H a : The population correlation coefficient IS significantly DIFFERENT FROM zero. There IS A SIGNIFICANT LINEAR RELATIONSHIP (correlation) between x and y in the population.

DRAWING A CONCLUSION: There are two methods of making the decision. The two methods are equivalent and give the same result.

• Method 1: Using the p -value
• Method 2: Using a table of critical values

In this chapter of this textbook, we will always use a significance level of 5%, α = 0.05

Using the p -value method, you could choose any appropriate significance level you want; you are not limited to using α = 0.05. But the table of critical values provided in this textbook assumes that we are using a significance level of 5%, α = 0.05. (If we wanted to use a different significance level than 5% with the critical value method, we would need different tables of critical values that are not provided in this textbook.)

## METHOD 1: Using a p -value to make a decision

To calculate the p -value using LinRegTTEST:

On the LinRegTTEST input screen, on the line prompt for β or ρ , highlight “ ≠ 0 “

The output screen shows the p-value on the line that reads “p =”.

• Decision: Reject the null hypothesis.
• Conclusion: “There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from zero.”
• Decision: DO NOT REJECT the null hypothesis.
• Conclusion: “There is insufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is NOT significantly different from zero.”
• You will use technology to calculate the p -value. The following describes the calculations to compute the test statistics and the p -value:
• The p -value is calculated using a t -distribution with n – 2 degrees of freedom.
• The formula for the test statistic is $$t=\frac{r\sqrt{n-2}}{\sqrt{1-{r}^{2}}}$$. The value of the test statistic, t , is shown in the computer or calculator output along with the p -value. The test statistic t has the same sign as the correlation coefficient r .
• The p -value is the combined area in both tails.

An alternative way to calculate the p -value (p) given by LinRegTTest is the command 2*tcdf(abs(t),10^99, n-2) in 2nd DISTR.

• Consider the third exam/final exam example .
• The line of best fit is: ŷ = -173.51 + 4.83 x with r = 0.6631 and there are n = 11 data points.
• Can the regression line be used for prediction? Given a third exam score ( x value), can we use the line to predict the final exam score (predicted y value)?
• H 0 : ρ = 0

H a : ρ ≠ 0

• The p -value is 0.026 (from LinRegTTest on your calculator or from computer software).
• The p -value, 0.026, is less than the significance level of α = 0.05.
• Decision: Reject the Null Hypothesis H 0
• Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score ( x ) and the final exam score ( y ) because the correlation coefficient is significantly different from zero.

Because r is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.

## METHOD 2: Using a table of Critical Values to make a decision

The 95% Critical Values of the Sample Correlation Coefficient Table can be used to give you a good idea of whether the computed value of $$r$$ is significant or not . Compare r to the appropriate critical value in the table. If r is not between the positive and negative critical values, then the correlation coefficient is significant. If r is significant, then you may want to use the line for prediction.

Suppose you computed r = 0.801 using n = 10 data points. df = n – 2 = 10 – 2 = 8. The critical values associated with df = 8 are -0.632 and + 0.632. If r < negative critical value or r > positive critical value, then r issignificant. Since r = 0.801 and 0.801 > 0.632, r is significant and the line may be usedfor prediction. If you view this example on a number line, it will help you.

For a given line of best fit, you computed that r = 0.6501 using n = 12 data points and the critical value is 0.576. Can the line be used for prediction? Why or why not?

If the scatter plot looks linear then, yes, the line can be used for prediction, because r > the positive critical value.

Suppose you computed r = –0.624 with 14 data points. df = 14 – 2 = 12. The critical values are –0.532 and 0.532. Since –0.624 < –0.532, r is significant and the line can be used for prediction

For a given line of best fit, you compute that r = 0.5204 using n = 9 data points, and the critical value is 0.666. Can the line be used for prediction? Why or why not?

No, the line cannot be used for prediction, because r < the positive critical value.

Suppose you computed r = 0.776 and n = 6. df = 6 – 2 = 4. The critical values are –0.811 and 0.811. Since –0.811 < 0.776 < 0.811, r is not significant, and the line should not be used for prediction.

For a given line of best fit, you compute that r = –0.7204 using n = 8 data points, and the critical value is = 0.707. Can the line be used for prediction? Why or why not?

Yes, the line can be used for prediction, because r < the negative critical value.

## THIRD-EXAM vs FINAL-EXAM EXAMPLE: critical value method

Consider the third exam/final exam example . The line of best fit is: ŷ = –173.51+4.83 x with r = 0.6631 and there are n = 11 data points. Can the regression line be used for prediction? Given a third-exam score ( x value), can we use the line to predict the final exam score (predicted y value)?

• Use the “95% Critical Value” table for r with df = n – 2 = 11 – 2 = 9.
• The critical values are –0.602 and +0.602
• Since 0.6631 > 0.602, r is significant.
• Conclusion:There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score ( x ) and the final exam score ( y ) because the correlation coefficient is significantly different from zero.

Suppose you computed the following correlation coefficients. Using the table at the end of the chapter, determine if r is significant and the line of best fit associated with each r can be used to predict a y value. If it helps, draw a number line.

• r = –0.567 and the sample size, n , is 19. The df = n – 2 = 17. The critical value is –0.456. –0.567 < –0.456 so r is significant.
• r = 0.708 and the sample size, n , is nine. The df = n – 2 = 7. The critical value is 0.666. 0.708 > 0.666 so r is significant.
• r = 0.134 and the sample size, n , is 14. The df = 14 – 2 = 12. The critical value is 0.532. 0.134 is between –0.532 and 0.532 so r is not significant.
• r = 0 and the sample size, n , is five. No matter what the dfs are, r = 0 is between the two critical values so r is not significant.

For a given line of best fit, you compute that r = 0 using n = 100 data points. Can the line be used for prediction? Why or why not?

No, the line cannot be used for prediction no matter what the sample size is.

## Assumptions in Testing the Significance of the Correlation Coefficient

Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between x and y in the sample data provides strong enough evidence so that we can conclude that there is a linear relationship between x and y in the population.

The regression line equation that we calculate from the sample data gives the best-fit line for our particular sample. We want to use this best-fit line for the sample as an estimate of the best-fit line for the population. Examining the scatterplot and testing the significance of the correlation coefficient helps us determine if it is appropriate to do this.

• There is a linear relationship in the population that models the average value of y for varying values of x . In other words, the expected value of y for each particular value lies on a straight line in the population. (We do not know the equation for the line for the population. Our regression line from the sample is our best estimate of this line in the population.)
• The y values for any particular x value are normally distributed about the line. This implies that there are more y values scattered closer to the line than are scattered farther away. Assumption (1) implies that these normal distributions are centered on the line: the means of these normal distributions of y values lie on the line.
• The standard deviations of the population y values about the line are equal for each value of x . In other words, each of these normal distributions of y values has the same shape and spread about the line.
• The residual errors are mutually independent (no pattern).
• The data are produced from a well-designed, random sample or randomized experiment.

## Chapter Review

Linear regression is a procedure for fitting a straight line of the form ŷ = a + bx to data. The conditions for regression are:

• Linear In the population, there is a linear relationship that models the average value of y for different values of x .
• Independent The residuals are assumed to be independent.
• Normal The y values are distributed normally for any value of x .
• Equal variance The standard deviation of the y values is equal for each x value.
• Random The data are produced from a well-designed random sample or randomized experiment.

The slope b and intercept a of the least-squares line estimate the slope β and intercept α of the population (true) regression line. To estimate the population standard deviation of y , σ , use the standard deviation of the residuals, s . $$s=\sqrt{\frac{SEE}{n-2}}$$. The variable ρ (rho) is the population correlation coefficient. To test the null hypothesis H 0 : ρ = hypothesized value , use a linear regression t-test. The most common null hypothesis is H 0 : ρ = 0 which indicates there is no linear relationship between x and y in the population. The TI-83, 83+, 84, 84+ calculator function LinRegTTest can perform this test (STATS TESTS LinRegTTest).

## Formula Review

Least Squares Line or Line of Best Fit:

$$\stackrel{^}{y}=a+bx$$

a = y -intercept

Standard deviation of the residuals:

$$s=\sqrt{\frac{SEE}{n-2}}.$$

SSE = sum of squared errors

n = the number of data points

When testing the significance of the correlation coefficient, what is the null hypothesis?

When testing the significance of the correlation coefficient, what is the alternative hypothesis?

If the level of significance is 0.05 and the p -value is 0.04, what conclusion can you draw?

If the level of significance is 0.05 and the p -value is 0.06, what conclusion can you draw?

We do not reject the null hypothesis. There is not sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is not significantly different from zero.

If there are 15 data points in a set of data, what is the number of degree of freedom?

## How to Perform a t-Test for Correlation

A Pearson correlation coefficient is used to quantify the linear association between two variables.

It always takes on a value between -1 and 1 where:

• -1 indicates a perfectly negative linear correlation.
• 0 indicates no linear correlation.
• 1 indicates a perfectly positive linear correlation.

To determine if a correlation coefficient is statistically significant you can perform a t-test, which involves calculating a t-score and a corresponding p-value.

The formula to calculate the t-score is:

• t = r√ (n-2) / (1-r 2 )
• r: The correlation coefficient
• n: The sample size

The p-value is calculated as the corresponding two-sided p-value for the t-distribution with n-2 degrees of freedom.

The following example shows how to perform a t-test for a correlation coefficient.

## Example: Performing a t-Test for Correlation

Suppose we have the following dataset with two variables:

Using some statistical software (Excel, R, Python, etc.) we can calculate the correlation coefficient between the two variables to be 0.707 .

This is a highly positive correlation, but to determine if it’s statistically significant we need to calculate the corresponding t-score and p-value.

We can calculate the t-score as:

• t = .707√ (10-2) / (1-.707 2 )

Using a T Score to P Value Calculator , we find that the corresponding p-value is 0.022 .

Since this p-value is less than .05, we would conclude that the correlation between these two variables is statistically significant.

How to Perform a Correlation Test in Excel How to Perform a Correlation Test in R What is Considered to Be a “Weak” Correlation? What is Considered to Be a “Strong” Correlation?

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## Chapter 12.5: Testing the Significance of the Correlation Coefficient

The correlation coefficient, r , tells us about the strength and direction of the linear relationship between x and y . However, the reliability of the linear model also depends on how many observed data points are in the sample. We need to look at both the value of the correlation coefficient r and the sample size n , together.

We perform a hypothesis test of the “significance of the correlation coefficient” to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population.

The sample data are used to compute r , the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But because we have only have sample data, we cannot calculate the population correlation coefficient. The sample correlation coefficient, r , is our estimate of the unknown population correlation coefficient.

• The symbol for the population correlation coefficient is ρ , the Greek letter “rho.”
• ρ = population correlation coefficient (unknown)
• r = sample correlation coefficient (known; calculated from sample data)

The hypothesis test lets us decide whether the value of the population correlation coefficient ρ is “close to zero” or “significantly different from zero”. We decide this based on the sample correlation coefficient r and the sample size n .

If the test concludes that the correlation coefficient is significantly different from zero, we say that the correlation coefficient is “significant.”

• Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from zero.
• What the conclusion means: There is a significant linear relationship between x and y . We can use the regression line to model the linear relationship between x and y in the population.

If the test concludes that the correlation coefficient is not significantly different from zero (it is close to zero), we say that correlation coefficient is “not significant”.

• Conclusion: “There is insufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is not significantly different from zero.”
• What the conclusion means: There is not a significant linear relationship between x and y . Therefore, we CANNOT use the regression line to model a linear relationship between x and y in the population.
• If r is significant and the scatter plot shows a linear trend, the line can be used to predict the value of y for values of x that are within the domain of observed x values.
• If r is not significant OR if the scatter plot does not show a linear trend, the line should not be used for prediction.
• If r is significant and if the scatter plot shows a linear trend, the line may NOT be appropriate or reliable for prediction OUTSIDE the domain of observed x values in the data.

## PERFORMING THE HYPOTHESIS TEST

• Null Hypothesis: H 0 : ρ = 0
• Alternate Hypothesis: H a : ρ ≠ 0

WHAT THE HYPOTHESES MEAN IN WORDS:

• Null Hypothesis H 0 : The population correlation coefficient IS NOT significantly different from zero. There IS NOT a significant linear relationship(correlation) between x and y in the population.
• Alternate Hypothesis H a : The population correlation coefficient IS significantly DIFFERENT FROM zero. There IS A SIGNIFICANT LINEAR RELATIONSHIP (correlation) between x and y in the population.

DRAWING A CONCLUSION: There are two methods of making the decision. The two methods are equivalent and give the same result.

• Method 1: Using the p -value
• Method 2: Using a table of critical values

In this chapter of this textbook, we will always use a significance level of 5%, α = 0.05

Using the p -value method, you could choose any appropriate significance level you want; you are not limited to using α = 0.05. But the table of critical values provided in this textbook assumes that we are using a significance level of 5%, α = 0.05. (If we wanted to use a different significance level than 5% with the critical value method, we would need different tables of critical values that are not provided in this textbook.)

## METHOD 1: Using a p -value to make a decision

To calculate the p -value using LinRegTTEST: On the LinRegTTEST input screen, on the line prompt for β or ρ , highlight “ ≠ 0 “ The output screen shows the p-value on the line that reads “p =”. (Most computer statistical software can calculate the p -value.)

• Decision: Reject the null hypothesis.
• Conclusion: “There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from zero.”
• Decision: DO NOT REJECT the null hypothesis.
• Conclusion: “There is insufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is NOT significantly different from zero.”
• You will use technology to calculate the p -value. The following describes the calculations to compute the test statistics and the p -value:
• The p -value is calculated using a t -distribution with n – 2 degrees of freedom.

• The p -value is the combined area in both tails.

An alternative way to calculate the p -value (p) given by LinRegTTest is the command 2*tcdf(abs(t),10^99, n-2) in 2nd DISTR.

• Consider the third exam/final exam example .
• The line of best fit is: ŷ = -173.51 + 4.83 x with r = 0.6631 and there are n = 11 data points.
• Can the regression line be used for prediction? Given a third exam score ( x value), can we use the line to predict the final exam score (predicted y value)?
• H 0 : ρ = 0

H a : ρ ≠ 0

• The p -value is 0.026 (from LinRegTTest on your calculator or from computer software).
• The p -value, 0.026, is less than the significance level of α = 0.05.
• Decision: Reject the Null Hypothesis H 0
• Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score ( x ) and the final exam score ( y ) because the correlation coefficient is significantly different from zero.

Because r is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.

## METHOD 2: Using a table of Critical Values to make a decision

Suppose you computed r = 0.801 using n = 10 data points. df = n – 2 = 10 – 2 = 8. The critical values associated with df = 8 are -0.632 and + 0.632. If r < negative critical value or r > positive critical value, then r issignificant. Since r = 0.801 and 0.801 > 0.632, r is significant and the line may be usedfor prediction. If you view this example on a number line, it will help you.

For a given line of best fit, you computed that r = 0.6501 using n = 12 data points and the critical value is 0.576. Can the line be used for prediction? Why or why not?

Suppose you computed r = –0.624 with 14 data points. df = 14 – 2 = 12. The critical values are –0.532 and 0.532. Since –0.624 < –0.532, r is significant and the line can be used for prediction

For a given line of best fit, you compute that r = 0.5204 using n = 9 data points, and the critical value is 0.666. Can the line be used for prediction? Why or why not?

Suppose you computed r = 0.776 and n = 6. df = 6 – 2 = 4. The critical values are –0.811 and 0.811. Since –0.811 < 0.776 < 0.811, r is not significant, and the line should not be used for prediction.

For a given line of best fit, you compute that r = –0.7204 using n = 8 data points, and the critical value is = 0.707. Can the line be used for prediction? Why or why not?

## THIRD-EXAM vs FINAL-EXAM EXAMPLE: critical value method

Consider the third exam/final exam example . The line of best fit is: ŷ = –173.51+4.83 x with r = 0.6631 and there are n = 11 data points. Can the regression line be used for prediction? Given a third-exam score ( x value), can we use the line to predict the final exam score (predicted y value)?

• Use the “95% Critical Value” table for r with df = n – 2 = 11 – 2 = 9.
• The critical values are –0.602 and +0.602
• Since 0.6631 > 0.602, r is significant.
• Conclusion:There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score ( x ) and the final exam score ( y ) because the correlation coefficient is significantly different from zero.

Suppose you computed the following correlation coefficients. Using the table at the end of the chapter, determine if r is significant and the line of best fit associated with each r can be used to predict a y value. If it helps, draw a number line.

• r = –0.567 and the sample size, n , is 19. The df = n – 2 = 17. The critical value is –0.456. –0.567 < –0.456 so r is significant.
• r = 0.708 and the sample size, n , is nine. The df = n – 2 = 7. The critical value is 0.666. 0.708 > 0.666 so r is significant.
• r = 0.134 and the sample size, n , is 14. The df = 14 – 2 = 12. The critical value is 0.532. 0.134 is between –0.532 and 0.532 so r is not significant.
• r = 0 and the sample size, n , is five. No matter what the dfs are, r = 0 is between the two critical values so r is not significant.

For a given line of best fit, you compute that r = 0 using n = 100 data points. Can the line be used for prediction? Why or why not?

## Assumptions in Testing the Significance of the Correlation Coefficient

Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between x and y in the sample data provides strong enough evidence so that we can conclude that there is a linear relationship between x and y in the population.

The regression line equation that we calculate from the sample data gives the best-fit line for our particular sample. We want to use this best-fit line for the sample as an estimate of the best-fit line for the population. Examining the scatterplot and testing the significance of the correlation coefficient helps us determine if it is appropriate to do this.

• There is a linear relationship in the population that models the average value of y for varying values of x . In other words, the expected value of y for each particular value lies on a straight line in the population. (We do not know the equation for the line for the population. Our regression line from the sample is our best estimate of this line in the population.)
• The y values for any particular x value are normally distributed about the line. This implies that there are more y values scattered closer to the line than are scattered farther away. Assumption (1) implies that these normal distributions are centered on the line: the means of these normal distributions of y values lie on the line.
• The standard deviations of the population y values about the line are equal for each value of x . In other words, each of these normal distributions of y values has the same shape and spread about the line.
• The residual errors are mutually independent (no pattern).
• The data are produced from a well-designed, random sample or randomized experiment.

## Chapter Review

Linear regression is a procedure for fitting a straight line of the form ŷ = a + bx to data. The conditions for regression are:

• Linear In the population, there is a linear relationship that models the average value of y for different values of x .
• Independent The residuals are assumed to be independent.
• Normal The y values are distributed normally for any value of x .
• Equal variance The standard deviation of the y values is equal for each x value.
• Random The data are produced from a well-designed random sample or randomized experiment.

## Formula Review

Least Squares Line or Line of Best Fit:

a = y -intercept

Standard deviation of the residuals:

SSE = sum of squared errors

n = the number of data points

When testing the significance of the correlation coefficient, what is the null hypothesis?

When testing the significance of the correlation coefficient, what is the alternative hypothesis?

If the level of significance is 0.05 and the p -value is 0.04, what conclusion can you draw?

## 12.3 Testing the Significance of the Correlation Coefficient (Optional)

The correlation coefficient, r , tells us about the strength and direction of the linear relationship between x and y . However, the reliability of the linear model also depends on how many observed data points are in the sample. We need to look at both the correlation coefficient r and the sample size n , together.

We perform a hypothesis test of the significance of the correlation coefficient to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population.

The sample data are used to compute r , the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But, because we have only sample data, we cannot calculate the population correlation coefficient. The sample correlation coefficient, r , is our estimate of the unknown population correlation coefficient.

• The symbol for the population correlation coefficient is ρ , the Greek letter rho.
• ρ = population correlation coefficient (unknown).
• r = sample correlation coefficient (known; calculated from sample data).

The hypothesis test lets us decide whether the value of the population correlation coefficient ρ is close to zero or significantly different from zero . We decide this based on the sample correlation coefficient r and the sample size n .

If the test concludes the correlation coefficient is significantly different from zero, we say the correlation coefficient is significant .

• Conclusion: There is sufficient evidence to conclude there is a significant linear relationship between x and y because the correlation coefficient is significantly different from zero.
• What the conclusion means: There is a significant linear relationship between x and y . We can use the regression line to model the linear relationship between x and y in the population.

If the test concludes the correlation coefficient is not significantly different from zero (it is close to zero), we say the correlation coefficient is not significant .

• Conclusion: There is insufficient evidence to conclude there is a significant linear relationship between x and y because the correlation coefficient is not significantly different from zero.
• What the conclusion means: There is not a significant linear relationship between x and y . Therefore, we cannot use the regression line to model a linear relationship between x and y in the population.
• If r is significant and the scatter plot shows a linear trend, the line can be used to predict the value of y for values of x that are within the domain of observed x values.
• If r is not significant or if the scatter plot does not show a linear trend, the line should not be used for prediction.
• If r is significant and the scatter plot shows a linear trend, the line may not be appropriate or reliable for prediction outside the domain of observed x values in the data.

## Performing the Hypothesis Test

• Null hypothesis: H 0 : ρ = 0.
• Alternate hypothesis: H a : ρ ≠ 0.

What the Hypothesis Means in Words:

• Null hypothesis H 0 : The population correlation coefficient is not significantly different from zero. There is not a significant linear relationship (correlation) between x and y in the population.
• Alternate hypothesis H a : The population correlation coefficient is significantly different from zero. There is a significant linear relationship (correlation) between x and y in the population.

Drawing a Conclusion: There are two methods to make a conclusion. The two methods are equivalent and give the same result.

• Method 1: Use the p -value.
• Method 2: Use a table of critical values.

In this chapter, we will always use a significance level of 5 percent, α = 0.05.

Using the p -value method, you could choose any appropriate significance level you want; you are not limited to using α = 0.05. But, the table of critical values provided in this textbook assumes we are using a significance level of 5 percent, α = 0.05. If we wanted to use a significance level different from 5 percent with the critical value method, we would need different tables of critical values that are not provided in this textbook.

## METHOD 1: Using a p -value to Make a Decision

Using the ti-83, 83+, 84, 84+ calculator.

To calculate the p -value using LinRegTTEST :

• Complete the same steps as the LinRegTTest performed previously in this chapter, making sure on the line prompt for β or σ , ≠ 0 is highlighted.
• When looking at the output screen, the p -value is on the line that reads p = .
• Decision: Reject the null hypothesis.
• Decision: Do not reject the null hypothesis.

You will use technology to calculate the p -value, but it is useful to know that the p -value is calculated using a t distribution with n – 2 degrees of freedom and that the p -value is the combined area in both tails.

An alternative way to calculate the p -value ( p ) given by LinRegTTest is the command 2*tcdf(abs(t),10^99, n–2) in 2nd DISTR.

• Consider the third exam/final exam example .
• The line of best fit is ŷ = –173.51 + 4.83 x , with r = 0.6631, and there are n = 11 data points.
• Can the regression line be used for prediction? Given a third exam score ( x value), can we use the line to predict the final exam score (predicted y value)?
• H 0 : ρ = 0
• H a : ρ ≠ 0
• The p -value is 0.026 (from LinRegTTest on a calculator or from computer software).
• The p -value, 0.026, is less than the significance level of α = 0.05.
• Decision: Reject the null hypothesis H 0 .
• Conclusion: There is sufficient evidence to conclude there is a significant linear relationship between the third exam score ( x ) and the final exam score ( y ) because the correlation coefficient is significantly different from zero.

Because r is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.

## METHOD 2: Using a Table of Critical Values to Make a Decision

The 95 Percent Critical Values of the Sample Correlation Coefficient Table ( Table 12.9 ) can be used to give you a good idea of whether the computed value of r is significant. Use it to find the critical values using the degrees of freedom, df = n – 2. The table has already been calculated with α = 0.05. The table tells you the positive critical value, but you should also make that number negative to have two critical values. If r is not between the positive and negative critical values, then the correlation coefficient is significant. If r is significant, then you may use the line for prediction. If r is not significant (between the critical values), you should not use the line to make predictions.

## Example 12.6

Suppose you computed r = 0.801 using n = 10 data points. The degrees of freedom would be 8 ( df = n – 2 = 10 – 2 = 8). Using Table 12.9 with df = 8, we find that the critical value is 0.632. This means the critical values are really ±0.632. Since r = 0.801 and 0.801 > 0.632, r is significant and the line may be used for prediction. If you view this example on a number line, it will help you to see that r is not between the two critical values.

## Try It 12.6

For a given line of best fit, you computed that r = 0.6501 using n = 12 data points, and the critical value found on the table is 0.576. Can the line be used for prediction? Why or why not?

## Example 12.7

Suppose you computed r = –0.624 with 14 data points, where df = 14 – 2 = 12. The critical values are –0.532 and 0.532. Since –0.624 < –0.532, r is significant and the line can be used for prediction.

## Try It 12.7

For a given line of best fit, you compute that r = 0.5204 using n = 9 data points, and the critical values are ±0.666. Can the line be used for prediction? Why or why not?

## Example 12.8

Suppose you computed r = 0.776 and n = 6, with df = 6 -– 2 = 4. The critical values are – 0.811 and 0.811. Since 0.776 is between the two critical values, r is not significant. The line should not be used for prediction.

## Try It 12.8

For a given line of best fit, you compute that r = –0.7204 using n = 8 data points, and the critical value is 0.707. Can the line be used for prediction? Why or why not?

## Third Exam vs. Final Exam Example: Critical Value Method

Consider the third exam/final exam example . The line of best fit is: ŷ = –173.51 + 4.83 x , with r = .6631, and there are n = 11 data points. Can the regression line be used for prediction? Given a third exam score ( x value), can we use the line to predict the final exam score (predicted y value)?

• Use the 95 Percent Critical Values table for r with df = n – 2 = 11 – 2 = 9.
• Using the table with df = 9, we find that the critical value listed is 0.602. Therefore, the critical values are ±0.602.
• Since 0.6631 > 0.602, r is significant.

## Example 12.9

Suppose you computed the following correlation coefficients. Using the table at the end of the chapter, determine whether r is significant and whether the line of best fit associated with each correlation coefficient can be used to predict a y value. If it helps, draw a number line.

• r = –0.567 and the sample size, n , is 19. To solve this problem, first find the degrees of freedom. df = n - 2 = 17. Then, using the table, the critical values are ±0.456. –0.567 < –0.456, or you may say that –0.567 is not between the two critical values. r is significant and may be used for predictions.
• r = 0.708 and the sample size, n , is 9. df = n - 2 = 7 The critical values are ±0.666. 0.708 > 0.666. r is significant and may be used for predictions.
• r = 0.134 and the sample size, n , is 14. df = 14 –- 2 = 12. The critical values are ±0.532. 0.134 is between –0.532 and 0.532. r is not significant and may not be used for predictions.
• r = 0 and the sample size, n , is 5. It doesn’'t matter what the degrees of freedom are because r = 0 will always be between the two critical values, so r is not significant and may not be used for predictions.

## Try It 12.9

For a given line of best fit, you compute that r = 0 using n = 100 data points. Can the line be used for prediction? Why or why not?

## Assumptions in Testing the Significance of the Correlation Coefficient

Testing the significance of the correlation coefficient requires that certain assumptions about the data be satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between x and y in the sample data provides strong enough evidence that we can conclude there is a linear relationship between x and y in the population.

The regression line equation that we calculate from the sample data gives the best-fit line for our particular sample. We want to use this best-fit line for the sample as an estimate of the best-fit line for the population. Examining the scatter plot and testing the significance of the correlation coefficient helps us determine whether it is appropriate to do this.

• There is a linear relationship in the population that models the sample data. Our regression line from the sample is our best estimate of this line in the population.
• The y values for any particular x value are normally distributed about the line. This implies there are more y values scattered closer to the line than are scattered farther away. Assumption 1 implies that these normal distributions are centered on the line; the means of these normal distributions of y values lie on the line.
• Normal distributions of all the y values have the same shape and spread about the line.
• The residual errors are mutually independent (no pattern).
• The data are produced from a well-designed, random sample or randomized experiment.

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## 1.9 - Hypothesis Test for the Population Correlation Coefficient

There is one more point we haven't stressed yet in our discussion about the correlation coefficient r and the coefficient of determination $$R^{2}$$ — namely, the two measures summarize the strength of a linear relationship in samples only . If we obtained a different sample, we would obtain different correlations, different $$R^{2}$$ values, and therefore potentially different conclusions. As always, we want to draw conclusions about populations , not just samples. To do so, we either have to conduct a hypothesis test or calculate a confidence interval. In this section, we learn how to conduct a hypothesis test for the population correlation coefficient $$\rho$$ (the greek letter "rho").

In general, a researcher should use the hypothesis test for the population correlation $$\rho$$ to learn of a linear association between two variables, when it isn't obvious which variable should be regarded as the response. Let's clarify this point with examples of two different research questions.

Consider evaluating whether or not a linear relationship exists between skin cancer mortality and latitude. We will see in Lesson 2 that we can perform either of the following tests:

• t -test for testing $$H_{0} \colon \beta_{1}= 0$$
• ANOVA F -test for testing $$H_{0} \colon \beta_{1}= 0$$

For this example, it is fairly obvious that latitude should be treated as the predictor variable and skin cancer mortality as the response.

By contrast, suppose we want to evaluate whether or not a linear relationship exists between a husband's age and his wife's age ( Husband and Wife data ). In this case, one could treat the husband's age as the response:

...or one could treat the wife's age as the response:

In cases such as these, we answer our research question concerning the existence of a linear relationship by using the t -test for testing the population correlation coefficient $$H_{0}\colon \rho = 0$$.

Let's jump right to it! We follow standard hypothesis test procedures in conducting a hypothesis test for the population correlation coefficient $$\rho$$.

## Steps for Hypothesis Testing for $$\boldsymbol{\rho}$$

Step 1: hypotheses.

First, we specify the null and alternative hypotheses:

• Null hypothesis $$H_{0} \colon \rho = 0$$
• Alternative hypothesis $$H_{A} \colon \rho ≠ 0$$ or $$H_{A} \colon \rho < 0$$ or $$H_{A} \colon \rho > 0$$

## Step 2: Test Statistic

Second, we calculate the value of the test statistic using the following formula:

Test statistic:  $$t^*=\dfrac{r\sqrt{n-2}}{\sqrt{1-R^2}}$$

## Step 3: P-Value

Third, we use the resulting test statistic to calculate the P -value. As always, the P -value is the answer to the question "how likely is it that we’d get a test statistic t* as extreme as we did if the null hypothesis were true?" The P -value is determined by referring to a t- distribution with n -2 degrees of freedom.

## Step 4: Decision

Finally, we make a decision:

• If the P -value is smaller than the significance level $$\alpha$$, we reject the null hypothesis in favor of the alternative. We conclude that "there is sufficient evidence at the$$\alpha$$ level to conclude that there is a linear relationship in the population between the predictor x and response y."
• If the P -value is larger than the significance level $$\alpha$$, we fail to reject the null hypothesis. We conclude "there is not enough evidence at the  $$\alpha$$ level to conclude that there is a linear relationship in the population between the predictor x and response y ."

## Example 1-5: Husband and Wife Data

Let's perform the hypothesis test on the husband's age and wife's age data in which the sample correlation based on n = 170 couples is r = 0.939. To test $$H_{0} \colon \rho = 0$$ against the alternative $$H_{A} \colon \rho ≠ 0$$, we obtain the following test statistic:

\begin{align} t^*&=\dfrac{r\sqrt{n-2}}{\sqrt{1-R^2}}\\ &=\dfrac{0.939\sqrt{170-2}}{\sqrt{1-0.939^2}}\\ &=35.39\end{align}

To obtain the P -value, we need to compare the test statistic to a t -distribution with 168 degrees of freedom (since 170 - 2 = 168). In particular, we need to find the probability that we'd observe a test statistic more extreme than 35.39, and then, since we're conducting a two-sided test, multiply the probability by 2. Minitab helps us out here:

## Student's t distribution with 168 DF

The output tells us that the probability of getting a test-statistic smaller than 35.39 is greater than 0.999. Therefore, the probability of getting a test-statistic greater than 35.39 is less than 0.001. As illustrated in the following video, we multiply by 2 and determine that the P-value is less than 0.002.

Since the P -value is small — smaller than 0.05, say — we can reject the null hypothesis. There is sufficient statistical evidence at the $$\alpha = 0.05$$ level to conclude that there is a significant linear relationship between a husband's age and his wife's age.

Incidentally, we can let statistical software like Minitab do all of the dirty work for us. In doing so, Minitab reports:

## Correlation: WAge, HAge

Pearson correlation of WAge and HAge = 0.939

P-Value = 0.000

One final note ... as always, we should clarify when it is okay to use the t -test for testing $$H_{0} \colon \rho = 0$$? The guidelines are a straightforward extension of the "LINE" assumptions made for the simple linear regression model. It's okay:

• When it is not obvious which variable is the response.
• For each x , the y 's are normal with equal variances.
• For each y , the x 's are normal with equal variances.
• Either, y can be considered a linear function of x .
• Or, x can be considered a linear function of y .
• The ( x , y ) pairs are independent

## Module 12: Linear Regression and Correlation

Testing the significance of the correlation coefficient, learning outcomes.

• Calculate and interpret the correlation coefficient

The correlation coefficient,  r , tells us about the strength and direction of the linear relationship between x and y . However, the reliability of the linear model also depends on how many observed data points are in the sample. We need to look at both the value of the correlation coefficient r and the sample size n , together.

We perform a hypothesis test of the “ significance of the correlation coefficient ” to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population.

The sample data are used to compute  r , the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But because we have only have sample data, we cannot calculate the population correlation coefficient. The sample correlation coefficient, r , is our estimate of the unknown population correlation coefficient.

• The symbol for the population correlation coefficient is ρ , the Greek letter “rho.”
• ρ = population correlation coefficient (unknown)
• r = sample correlation coefficient (known; calculated from sample data)

The hypothesis test lets us decide whether the value of the population correlation coefficient ρ is “close to zero” or “significantly different from zero”. We decide this based on the sample correlation coefficient r and the sample size n .

If the test concludes that the correlation coefficient is significantly different from zero, we say that the correlation coefficient is “significant.” Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from zero. What the conclusion means: There is a significant linear relationship between x and y . We can use the regression line to model the linear relationship between x and y in the population.

If the test concludes that the correlation coefficient is not significantly different from zero (it is close to zero), we say that correlation coefficient is “not significant.”

Conclusion: “There is insufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is not significantly different from zero.” What the conclusion means: There is not a significant linear relationship between x and y . Therefore, we CANNOT use the regression line to model a linear relationship between x and y in the population.

• If r is significant and the scatter plot shows a linear trend, the line can be used to predict the value of y for values of x that are within the domain of observed x values.
• If r is not significant OR if the scatter plot does not show a linear trend, the line should not be used for prediction.
• If r is significant and if the scatter plot shows a linear trend, the line may NOT be appropriate or reliable for prediction OUTSIDE the domain of observed x values in the data.

## Performing the Hypothesis Test

• Null Hypothesis: H 0 : ρ = 0
• Alternate Hypothesis: H a : ρ ≠ 0

## What the Hypotheses Mean in Words

• Null Hypothesis H 0 : The population correlation coefficient IS NOT significantly different from zero. There IS NOT a significant linear relationship(correlation) between x and y in the population.
• Alternate Hypothesis H a : The population correlation coefficient IS significantly DIFFERENT FROM zero. There IS A SIGNIFICANT LINEAR RELATIONSHIP (correlation) between x and y in the population.

## Drawing a Conclusion

There are two methods of making the decision. The two methods are equivalent and give the same result.

• Method 1: Using the p -value
• Method 2: Using a table of critical values

In this chapter of this textbook, we will always use a significance level of 5%,  α = 0.05

Using the  p -value method, you could choose any appropriate significance level you want; you are not limited to using α = 0.05. But the table of critical values provided in this textbook assumes that we are using a significance level of 5%, α = 0.05. (If we wanted to use a different significance level than 5% with the critical value method, we would need different tables of critical values that are not provided in this textbook.)

## Method 1: Using a p -value to make a decision

To calculate the  p -value using LinRegTTEST:

• On the LinRegTTEST input screen, on the line prompt for β or ρ , highlight “≠ 0”
• The output screen shows the p-value on the line that reads “p =”.
• (Most computer statistical software can calculate the p -value.)

If the p -value is less than the significance level ( α = 0.05)

• Decision: Reject the null hypothesis.
• Conclusion: “There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from zero.”

If the p -value is NOT less than the significance level ( α = 0.05)

• Decision: DO NOT REJECT the null hypothesis.
• Conclusion: “There is insufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is NOT significantly different from zero.”

Calculation Notes:

• You will use technology to calculate the p -value. The following describes the calculations to compute the test statistics and the p -value:
• The p -value is calculated using a t -distribution with n – 2 degrees of freedom.
• The formula for the test statistic is $\displaystyle{t}=\frac{{{r}\sqrt{{{n}-{2}}}}}{\sqrt{{{1}-{r}^{{2}}}}}$. The value of the test statistic, t , is shown in the computer or calculator output along with the p -value. The test statistic t has the same sign as the correlation coefficient r .
• The p -value is the combined area in both tails.

An alternative way to calculate the  p -value (p) given by LinRegTTest is the command 2*tcdf(abs(t),10^99, n-2) in 2nd DISTR.

## Method 2: Using a table of Critical Values to make a decision

The 95% Critical Values of the Sample Correlation Coefficient Table can be used to give you a good idea of whether the computed value of is significant or not. Compare  r to the appropriate critical value in the table. If r is not between the positive and negative critical values, then the correlation coefficient is significant. If r is significant, then you may want to use the line for prediction.

Suppose you computed  r = 0.801 using n = 10 data points. df = n – 2 = 10 – 2 = 8. The critical values associated with df = 8 are -0.632 and + 0.632. If r < negative critical value or r > positive critical value, then r is  significant . Since r = 0.801 and 0.801 > 0.632, r is significant and the line may be used for prediction. If you view this example on a number line, it will help you.

For a given line of best fit, you computed that  r = 0.6501 using n = 12 data points and the critical value is 0.576. Can the line be used for prediction? Why or why not?

If the scatter plot looks linear then, yes, the line can be used for prediction, because  r > the positive critical value.

Suppose you computed  r = –0.624 with 14 data points. df = 14 – 2 = 12. The critical values are –0.532 and 0.532. Since –0.624 < –0.532, r is significant and the line can be used for prediction

For a given line of best fit, you compute that  r = 0.5204 using n = 9 data points, and the critical value is 0.666. Can the line be used for prediction? Why or why not?

No, the line cannot be used for prediction, because  r < the positive critical value.

Suppose you computed  r = 0.776 and n = 6. df = 6 – 2 = 4. The critical values are –0.811 and 0.811. Since –0.811 < 0.776 < 0.811, r is not significant, and the line should not be used for prediction.

–0.811 <  r = 0.776 < 0.811. Therefore, r is not significant.

For a given line of best fit, you compute that  r = –0.7204 using n = 8 data points, and the critical value is = 0.707. Can the line be used for prediction? Why or why not?

Yes, the line can be used for prediction, because  r < the negative critical value.

Suppose you computed the following correlation coefficients. Using the table at the end of the chapter, determine if  r is significant and the line of best fit associated with each r can be used to predict a y value. If it helps, draw a number line.

• r = –0.567 and the sample size, n , is 19. The df = n – 2 = 17. The critical value is –0.456. –0.567 < –0.456 so r is significant.
• r = 0.708 and the sample size, n , is nine. The df = n – 2 = 7. The critical value is 0.666. 0.708 > 0.666 so r is significant.
• r = 0.134 and the sample size, n , is 14. The df = 14 – 2 = 12. The critical value is 0.532. 0.134 is between –0.532 and 0.532 so r is not significant.
• r = 0 and the sample size, n , is five. No matter what the dfs are, r = 0 is between the two critical values so r is not significant.

For a given line of best fit, you compute that  r = 0 using n = 100 data points. Can the line be used for prediction? Why or why not?

No, the line cannot be used for prediction no matter what the sample size is.

## Assumptions in Testing the Significance of the Correlation Coefficient

Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between x and y in the sample data provides strong enough evidence so that we can conclude that there is a linear relationship between x and y in the population.

The regression line equation that we calculate from the sample data gives the best-fit line for our particular sample. We want to use this best-fit line for the sample as an estimate of the best-fit line for the population. Examining the scatterplot and testing the significance of the correlation coefficient helps us determine if it is appropriate to do this.

The assumptions underlying the test of significance are:

• There is a linear relationship in the population that models the average value of y for varying values of x . In other words, the expected value of y for each particular value lies on a straight line in the population. (We do not know the equation for the line for the population. Our regression line from the sample is our best estimate of this line in the population.)
• The y values for any particular x value are normally distributed about the line. This implies that there are more y values scattered closer to the line than are scattered farther away. Assumption (1) implies that these normal distributions are centered on the line: the means of these normal distributions of y values lie on the line.
• The standard deviations of the population y values about the line are equal for each value of x . In other words, each of these normal distributions of y values has the same shape and spread about the line.
• The residual errors are mutually independent (no pattern).
• The data are produced from a well-designed, random sample or randomized experiment.

The  y values for each x value are normally distributed about the line with the same standard deviation. For each x value, the mean of the y values lies on the regression line. More y values lie near the line than are scattered further away from the line.

## Concept Review

Linear regression is a procedure for fitting a straight line of the form $\displaystyle\hat{{y}}={a}+{b}{x}$ to data. The conditions for regression are:

• Linear: In the population, there is a linear relationship that models the average value of y for different values of x .
• Independent: The residuals are assumed to be independent.
• Normal: The y values are distributed normally for any value of x .
• Equal variance: The standard deviation of the y values is equal for each x value.
• Random: The data are produced from a well-designed random sample or randomized experiment.

The slope  b and intercept a of the least-squares line estimate the slope β and intercept α of the population (true) regression line. To estimate the population standard deviation of y , σ , use the standard deviation of the residuals, s .

$\displaystyle{s}=\sqrt{{\frac{{{S}{S}{E}}}{{{n}-{2}}}}}$ The variable ρ (rho) is the population correlation coefficient.

To test the null hypothesis  H 0 : ρ = hypothesized value , use a linear regression t-test. The most common null hypothesis is H 0 : ρ = 0 which indicates there is no linear relationship between x and y in the population.

The TI-83, 83+, 84, 84+ calculator function LinRegTTest can perform this test (STATS TESTS LinRegTTest).

## Formula Review

Least Squares Line or Line of Best Fit: $\displaystyle\hat{{y}}={a}+{b}{x}$

where  a = y -intercept,  b = slope

Standard deviation of the residuals:

$\displaystyle{s}=\sqrt{{\frac{{{S}{S}{E}}}{{{n}-{2}}}}}$

SSE = sum of squared errors

n = the number of data points

• OpenStax, Statistics, Testing the Significance of the Correlation Coefficient. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected]:83/Introductory_Statistics . License : CC BY: Attribution

## Correlation Calculator

Input your values with a space or comma between in the table below

## Critical Value

Results shown here

## Sample size, n

Sample correlation coefficient, r, standardized sample score.

## Correlation Coefficient Significance Calculator using p-value

Instructions: Use this Correlation Coefficient Significance Calculator to enter the sample correlation $$r$$, sample size $$n$$ and the significance level $$\alpha$$, and the solver will test whether or not the correlation coefficient is significantly different from zero using the critical correlation approach.

## More About Significance of the Correlation Coefficient Significance Calculator

The sample correlation $$r$$ is a statistic that estimates the population correlation, $$\rho$$. On typical statistical test consists of assessing whether or not the correlation coefficient is significantly different from zero.

There are least two methods to assess the significance of the sample correlation coefficient: One of them is based on the critical correlation. Such approach is based upon on the idea that if the sample correlation $$r$$ is large enough, then the population correlation $$\rho$$ is different from zero.

In order to assess whether or not the sample correlation is significantly different from zero, the following t-statistic is obtained

So, this is the formula for the t test for correlation coefficient, which the calculator will provide for you showing all the steps of the calculation.

If the above t-statistic is significant, then we would reject the null hypothesis $$H_0$$ (that the population correlation is zero). You can also the critical correlation approach , with the same purpose of assessing whether or not the sample correlation is significantly different from zero, but in that case by comparing the sample correlation with a critical correlation value.

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## 12.2.1: Hypothesis Test for Linear Regression

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• Rachel Webb
• Portland State University

To test to see if the slope is significant we will be doing a two-tailed test with hypotheses. The population least squares regression line would be $$y = \beta_{0} + \beta_{1} + \varepsilon$$ where $$\beta_{0}$$ (pronounced “beta-naught”) is the population $$y$$-intercept, $$\beta_{1}$$ (pronounced “beta-one”) is the population slope and $$\varepsilon$$ is called the error term.

If the slope were horizontal (equal to zero), the regression line would give the same $$y$$-value for every input of $$x$$ and would be of no use. If there is a statistically significant linear relationship then the slope needs to be different from zero. We will only do the two-tailed test, but the same rules for hypothesis testing apply for a one-tailed test.

We will only be using the two-tailed test for a population slope.

The hypotheses are:

$$H_{0}: \beta_{1} = 0$$ $$H_{1}: \beta_{1} \neq 0$$

The null hypothesis of a two-tailed test states that there is not a linear relationship between $$x$$ and $$y$$. The alternative hypothesis of a two-tailed test states that there is a significant linear relationship between $$x$$ and $$y$$.

Either a t-test or an F-test may be used to see if the slope is significantly different from zero. The population of the variable $$y$$ must be normally distributed.

## F-Test for Regression

An F-test can be used instead of a t-test. Both tests will yield the same results, so it is a matter of preference and what technology is available. Figure 12-12 is a template for a regression ANOVA table,

where $$n$$ is the number of pairs in the sample and $$p$$ is the number of predictor (independent) variables; for now this is just $$p = 1$$. Use the F-distribution with degrees of freedom for regression = $$df_{R} = p$$, and degrees of freedom for error = $$df_{E} = n - p - 1$$. This F-test is always a right-tailed test since ANOVA is testing the variation in the regression model is larger than the variation in the error.

Use an F-test to see if there is a significant relationship between hours studied and grade on the exam. Use $$\alpha$$ = 0.05.

## T-Test for Regression

If the regression equation has a slope of zero, then every $$x$$ value will give the same $$y$$ value and the regression equation would be useless for prediction. We should perform a t-test to see if the slope is significantly different from zero before using the regression equation for prediction. The numeric value of t will be the same as the t-test for a correlation. The two test statistic formulas are algebraically equal; however, the formulas are different and we use a different parameter in the hypotheses.

The formula for the t-test statistic is $$t = \frac{b_{1}}{\sqrt{ \left(\frac{MSE}{SS_{xx}}\right) }}$$

Use the t-distribution with degrees of freedom equal to $$n - p - 1$$.

The t-test for slope has the same hypotheses as the F-test:

Use a t-test to see if there is a significant relationship between hours studied and grade on the exam, use $$\alpha$$ = 0.05.

#### IMAGES

1. T-Test for a Correlation Coefficient and the Coefficient of Determination

2. PPT

3. Correlation Formula

4. Hypothesis Testing a Coefficient of Correlation

5. Hypothesis testing for correlation coefficient r

6. Correlation and regression

#### VIDEO

1. Correlation hypothesis testing part 1

2. Pearson Test for Correlation

3. Conduct a Linear Correlation Hypothesis Test Using Free Web Calculators

4. Testing of hypothesis about correlation coefficient

5. Conduct a Multiple Linear Correlation Hypothesis Test Using Free Web Calculators

6. Hypothesis Testing based on Correlation

1. 11.2: Correlation Hypothesis Test

The formula for the test statistic is t = r√n − 2 √1 − r2. t = r n − 2 √ 1 − r 2 √. The value of the test statistic, t. t. , is shown in the computer or calculator output along with the p-value. p -value. The test statistic t. t. has the same sign as the correlation coefficient r.

2. 1.9

Let's perform the hypothesis test on the husband's age and wife's age data in which the sample correlation based on n = 170 couples is r = 0.939. To test H 0: ρ = 0 against the alternative H A: ρ ≠ 0, we obtain the following test statistic: t ∗ = r n − 2 1 − R 2 = 0.939 170 − 2 1 − 0.939 2 = 35.39. To obtain the P -value, we need ...

3. 12.1.2: Hypothesis Test for a Correlation

The t-test is a statistical test for the correlation coefficient. It can be used when x x and y y are linearly related, the variables are random variables, and when the population of the variable y y is normally distributed. The formula for the t-test statistic is t = r ( n − 2 1 −r2)− −−−−−−−√ t = r ( n − 2 1 − r 2).

4. 12.4 Testing the Significance of the Correlation Coefficient

PERFORMING THE HYPOTHESIS TEST. Null Hypothesis: H 0: ρ = 0 Alternate Hypothesis: H a: ρ ≠ 0 WHAT THE HYPOTHESES MEAN IN WORDS: Null Hypothesis H 0: The population correlation coefficient IS NOT significantly different from zero. There IS NOT a significant linear relationship (correlation) between x and y in the population.; Alternate Hypothesis H a: The population correlation coefficient ...

5. Testing the Significance of the Correlation Coefficient

The hypothesis test lets us decide whether the value of the population correlation coefficient ρ is "close to zero" or "significantly different from zero". We decide this based on the sample correlation coefficient r and the sample size n .

6. 13.2 Testing the Significance of the Correlation Coefficient

The correlation coefficient, r, tells us about the strength and direction of the linear relationship between X 1 and X 2. The sample data are used to compute r, the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. ... The hypothesis test lets us decide ...

7. 9.4.1

There are 28 observations. The test statistic is: t ∗ = r n − 2 1 − r 2 = ( 0.711) 28 − 2 1 − 0.711 2 = 5.1556. Next, we need to find the p-value. The p-value for the two-sided test is: p-value = 2 P ( T > 5.1556) < 0.0001. Therefore, for any reasonable α level, we can reject the hypothesis that the population correlation coefficient ...

8. Conducting a Hypothesis Test for the Population Correlation Coefficient

We follow standard hypothesis test procedures in conducting a hypothesis test for the population correlation coefficient ρ. First, we specify the null and alternative hypotheses: Null hypothesis H0: ρ = 0. Alternative hypothesis HA: ρ ≠ 0 or HA: ρ < 0 or HA: ρ > 0. Second, we calculate the value of the test statistic using the following ...

9. Correlation Coefficient

i. = the difference between the x-variable rank and the y-variable rank for each pair of data. ∑ d2. i. = sum of the squared differences between x- and y-variable ranks. n = sample size. If you have a correlation coefficient of 1, all of the rankings for each variable match up for every data pair.

10. Pearson Correlation Coefficient (r)

Revised on February 10, 2024. The Pearson correlation coefficient (r) is the most common way of measuring a linear correlation. It is a number between -1 and 1 that measures the strength and direction of the relationship between two variables. When one variable changes, the other variable changes in the same direction.

11. How to Perform a t-Test for Correlation

To determine if a correlation coefficient is statistically significant you can perform a t-test, which involves calculating a t-score and a corresponding p-value. The formula to calculate the t-score is: t = r√(n-2) / (1-r2) where: The p-value is calculated as the corresponding two-sided p-value for the t-distribution with n-2 degrees of freedom.

12. Hypothesis Test for Correlation

The hypothesis test lets us decide whether the value of the population correlation coefficient ρ is "close to zero" or "significantly different from zero.". We decide this based on the sample correlation coefficient r and the sample size n. If the test concludes that the correlation coefficient is significantly different from zero, we ...

13. Chapter 12.5: Testing the Significance of the Correlation Coefficient

The hypothesis test lets us decide whether the value of the population correlation coefficient ρ is "close to zero" or "significantly different from zero". We decide this based on the sample correlation coefficient r and the sample size n .

14. 12.3 Testing the Significance of the Correlation Coefficient (Optional

Performing the Hypothesis Test. Null hypothesis: H 0: ρ = 0. Alternate hypothesis: H a: ρ ≠ 0. What the Hypothesis Means in Words: Null hypothesis H 0: The population correlation coefficient is not significantly different from zero. There is not a significant linear relationship (correlation) between x and y in the population.; Alternate hypothesis H a: The population correlation ...

15. Interpreting Correlation Coefficients

That description matches our moderate correlation coefficient of 0.694. For the hypothesis test, our p-value equals 0.000. This p-value is less than any reasonable significance level. ... On the other hand, the hypothesis test of Pearson's correlation coefficient does assume that the data follow a bivariate normal distribution. If you want to ...

16. Hypothesis Testing: Correlations

We perform a hypothesis test of the "significance of the correlation coefficient" to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population. The hypothesis test lets us decide whether the value of the population correlation coefficient. \rho ρ.

17. 1.9

In general, a researcher should use the hypothesis test for the population correlation \ (\rho\) to learn of a linear association between two variables, when it isn't obvious which variable should be regarded as the response. Let's clarify this point with examples of two different research questions. Consider evaluating whether or not a linear ...

18. Testing the Significance of the Correlation Coefficient

The correlation coefficient, r, tells us about the strength and direction of the linear relationship between x and y.However, the reliability of the linear model also depends on how many observed data points are in the sample. We need to look at both the value of the correlation coefficient r and the sample size n, together.. We perform a hypothesis test of the "significance of the ...

19. 2.5.2 Hypothesis Testing for Correlation

It is also possible to use a hypothesis test to determine whether a given product moment correlation coefficient calculated from a sample could be representative of the same relationship existing within the whole population. For full information on hypothesis testing, see the revision notes from section 5.1.1 Hypothesis Testing

20. Correlation Hypothesis Test Calculator for r

Discover the power of statistics with our free hypothesis test for Pearson correlation coefficient (r) on two numerical data sets. Our user-friendly calculator provides accurate results to determine the strength and significance of relationships between variables. Uncover valuable insights from your data and make informed decisions with ease.

21. Correlation Coefficient Significance Calculator using p-value

t = r\sqrt { \frac {n-2} {1-r^2}} t = r 1 −r2n −2. So, this is the formula for the t test for correlation coefficient, which the calculator will provide for you showing all the steps of the calculation. If the above t-statistic is significant, then we would reject the null hypothesis H_0 H 0 (that the population correlation is zero). You ...

22. 12.5: Testing the Significance of the Correlation Coefficient

The formula for the test statistic is t = r√n − 2 √1 − r2. The value of the test statistic, t, is shown in the computer or calculator output along with the p-value. The test statistic t has the same sign as the correlation coefficient r. The p-value is the combined area in both tails.

23. 12.2.1: Hypothesis Test for Linear Regression

The hypotheses are: Find the critical value using dfE = n − p − 1 = 13 for a two-tailed test α = 0.05 inverse t-distribution to get the critical values ± 2.160. Draw the sampling distribution and label the critical values, as shown in Figure 12-14. Figure 12-14: Graph of t-distribution with labeled critical values.

24. 2.5.2 Hypothesis Testing for Correlation

You should be familiar with using a hypothesis test to determine bias within probability problems. It is also possible to use a hypothesis test to determine whether a given product moment correlation coefficient calculated from a sample could be representative of the same relationship existing within the whole population. For full information on hypothesis testing, see the revision notes from ...

25. High Correlation Coefficient: Data Analytics Implications

A high correlation coefficient might tempt you to infer a causal relationship between the variables, but this is a common pitfall in data analytics. Just because two variables are correlated does ...