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Statistical Hypotheses and Error

• e.g., there is no link between disease and risk factor
• e.g., there is a link between disease and risk factor
• incorrectly rejecting null hypothesis
• α = probability of type I error
• general rule of thumb is that statistical significance is reached if p
• incorrectly accepting null hypothesis
• β = probability of type II error
• power = 1 - β
• increasing sample size increases power
• increasing effect size increases power
• Probability of correctly accepting null hypothesis
• usually done with 95% confidence interval (2 standard deviations from the mean)
• e.g., based on our study data, we are 95% confident that the average salary of a teacher lies between $30,000-45,000/year
• Confidence interval is calculated from statistics generated from the studied data
• Smaller confidence intervals suggest better precision of the data
• Larger confidence intervals suggest less precision of the data
• If confidence intervals of 2 groups overlap , there is no statistically significant difference
• comparisons planned prior to data analysis
• planning dependent on knowledge researchers have prior to conducting statistical tests
• researcher decides additional comparisons to make after viewing data
• post hoc analysis would involve comparing group A to group B, B to C, and A to C to see between which groups the difference lies
• one potential hazard is an increased likelihood of spurious statistical associations
• - Statistical Hypotheses and Error

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Biostatistics

Biostatistics: a usmle guide, introduction.

Biostatistics is a vital field within the medical sciences that encompasses the application of statistical methods to analyze and interpret data related to health and medicine. As a medical professional, understanding and effectively utilizing biostatistics is essential for clinical decision-making, research, and evidence-based practice. This USMLE guide aims to provide a comprehensive overview of the key concepts and principles of biostatistics to help you excel in your medical examinations and beyond.

Table of Contents

Variables and data types, descriptive statistics, probability, hypothesis testing, confidence intervals, regression and correlation.

• Epidemiology
• Clinical Trials
• Biostatistics in Research

Study Design

Understanding different study designs is crucial in interpreting medical research findings. The common study designs include:

• Observational Studies: These studies observe and record data without any intervention or manipulation of variables. Examples include cohort studies, case-control studies, and cross-sectional studies.
• Experimental Studies: These studies involve intervention or manipulation of variables to determine cause-and-effect relationships. Randomized controlled trials (RCTs) are the gold standard for experimental studies.
• Meta-Analyses: These studies pool and analyze data from multiple studies to derive conclusions with greater statistical power.

In biostatistics, variables are characteristics or attributes that can be measured or categorized. They can be classified into:

• Categorical Variables: These variables represent qualitative characteristics and are further divided into nominal (non-ordered categories) and ordinal (ordered categories) types.
• Continuous Variables: These variables represent quantitative measurements and can take any value within a range.

Descriptive statistics summarizes and describes the main features of a dataset. Common measures used in descriptive statistics include:

• Measures of Central Tendency: These measures include mean, median, and mode, which represent the center or average value of a dataset.
• Measures of Dispersion: These measures include range, variance, and standard deviation, which represent the spread or variability of a dataset.
• Percentiles and Quartiles: These measures divide the dataset into equal parts, providing insights into the distribution of data.

Probability is the mathematical framework for quantifying uncertainty. Key concepts in probability theory include:

• Probability Distribution: A function that describes the likelihood of different outcomes in a sample space.
• Random Variables: Variables whose outcomes are determined by chance.
• Conditional Probability: The probability of an event occurring given that another event has already occurred.
• Bayes' Theorem: A fundamental theorem used to update the probability of an event based on new information.

Hypothesis testing is a statistical method to evaluate the validity of a claim or hypothesis about a population parameter. The process involves:

• Null Hypothesis (H0): A statement of no effect or no difference.
• Alternative Hypothesis (Ha): A statement that contradicts the null hypothesis.
• p-value: The probability of obtaining the observed data or more extreme results assuming the null hypothesis is true.
• Type I and Type II Errors: Errors that can occur in hypothesis testing and their implications.

Confidence intervals provide a range of values within which a population parameter is likely to fall. Key points about confidence intervals include:

• Margin of Error: The range around the point estimate that defines the confidence interval.
• Confidence Level: The level of confidence (expressed as a percentage) that the true population parameter lies within the calculated interval.
• Interpreting Confidence Intervals: If the interval includes the null value, it suggests that the difference is not statistically significant.

Regression and correlation analysis are used to understand relationships between variables. Key concepts include:

• Linear Regression: A statistical technique to model the relationship between a dependent variable and one or more independent variables.
• Correlation Coefficient: A measure of the strength and direction of the linear relationship between two variables (ranging from

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9.1 Null and Alternative Hypotheses

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 , the — null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

H a —, the alternative hypothesis: a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are reject H 0 if the sample information favors the alternative hypothesis or do not reject H 0 or decline to reject H 0 if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example 9.1

H 0 : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30 H a : More than 30 percent of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25 percent. State the null and alternative hypotheses.

Example 9.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are the following: H 0 : μ = 2.0 H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 66
• H a : μ __ 66

Example 9.3

We want to test if college students take fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following: H 0 : μ ≥ 5 H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 45
• H a : μ __ 45

Example 9.4

An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066

On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : p __ 0.40
• H a : p __ 0.40

Collaborative Exercise

Bring to class a newspaper, some news magazines, and some internet articles. In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

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Module 9: Hypothesis Testing With One Sample

Null and alternative hypotheses, learning outcomes.

• Describe hypothesis testing in general and in practice

The actual test begins by considering two  hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 : The null hypothesis: It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.

H a : The alternative hypothesis : It is a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make adecision. There are two options for a  decision . They are “reject H 0 ” if the sample information favors the alternative hypothesis or “do not reject H 0 ” or “decline to reject H 0 ” if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in  H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

H 0 : No more than 30% of the registered voters in Santa Clara County voted in the primary election. p ≤ 30

H a : More than 30% of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.

H 0 : The drug reduces cholesterol by 25%. p = 0.25

H a : The drug does not reduce cholesterol by 25%. p ≠ 0.25

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are:

H 0 : μ = 2.0

H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : μ __ 66 H a : μ __ 66

• H 0 : μ = 66
• H a : μ ≠ 66

We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are:

H 0 : μ ≥ 5

H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : μ __ 45 H a : μ __ 45

• H 0 : μ ≥ 45
• H a : μ < 45

In an issue of U.S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses.

H 0 : p ≤ 0.066

H a : p > 0.066

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses. H 0 : p __ 0.40 H a : p __ 0.40

• H 0 : p = 0.40
• H a : p > 0.40

Concept Review

In a  hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we: Evaluate the null hypothesis , typically denoted with H 0 . The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality (=, ≤ or ≥) Always write the alternative hypothesis , typically denoted with H a or H 1 , using less than, greater than, or not equals symbols, i.e., (≠, >, or <). If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis. Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.

Formula Review

H 0 and H a are contradictory.

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10.1 - setting the hypotheses: examples.

A significance test examines whether the null hypothesis provides a plausible explanation of the data. The null hypothesis itself does not involve the data. It is a statement about a parameter (a numerical characteristic of the population). These population values might be proportions or means or differences between means or proportions or correlations or odds ratios or any other numerical summary of the population. The alternative hypothesis is typically the research hypothesis of interest. Here are some examples.

Example 10.2: Hypotheses with One Sample of One Categorical Variable Section  

About 10% of the human population is left-handed. Suppose a researcher at Penn State speculates that students in the College of Arts and Architecture are more likely to be left-handed than people found in the general population. We only have one sample since we will be comparing a population proportion based on a sample value to a known population value.

• Research Question : Are artists more likely to be left-handed than people found in the general population?
• Response Variable : Classification of the student as either right-handed or left-handed

State Null and Alternative Hypotheses

• Null Hypothesis : Students in the College of Arts and Architecture are no more likely to be left-handed than people in the general population (population percent of left-handed students in the College of Art and Architecture = 10% or p = .10).
• Alternative Hypothesis : Students in the College of Arts and Architecture are more likely to be left-handed than people in the general population (population percent of left-handed students in the College of Arts and Architecture > 10% or p > .10). This is a one-sided alternative hypothesis.

Example 10.3: Hypotheses with One Sample of One Measurement Variable Section  

A generic brand of the anti-histamine Diphenhydramine markets a capsule with a 50 milligram dose. The manufacturer is worried that the machine that fills the capsules has come out of calibration and is no longer creating capsules with the appropriate dosage.

• Research Question : Does the data suggest that the population mean dosage of this brand is different than 50 mg?
• Response Variable : dosage of the active ingredient found by a chemical assay.
• Null Hypothesis : On the average, the dosage sold under this brand is 50 mg (population mean dosage = 50 mg).
• Alternative Hypothesis : On the average, the dosage sold under this brand is not 50 mg (population mean dosage ≠ 50 mg). This is a two-sided alternative hypothesis.

Example 10.4: Hypotheses with Two Samples of One Categorical Variable Section  

Many people are starting to prefer vegetarian meals on a regular basis. Specifically, a researcher believes that females are more likely than males to eat vegetarian meals on a regular basis.

• Research Question : Does the data suggest that females are more likely than males to eat vegetarian meals on a regular basis?
• Response Variable : Classification of whether or not a person eats vegetarian meals on a regular basis
• Explanatory (Grouping) Variable: Sex
• Null Hypothesis : There is no sex effect regarding those who eat vegetarian meals on a regular basis (population percent of females who eat vegetarian meals on a regular basis = population percent of males who eat vegetarian meals on a regular basis or p females = p males ).
• Alternative Hypothesis : Females are more likely than males to eat vegetarian meals on a regular basis (population percent of females who eat vegetarian meals on a regular basis > population percent of males who eat vegetarian meals on a regular basis or p females > p males ). This is a one-sided alternative hypothesis.

Example 10.5: Hypotheses with Two Samples of One Measurement Variable Section  

Obesity is a major health problem today. Research is starting to show that people may be able to lose more weight on a low carbohydrate diet than on a low fat diet.

• Research Question : Does the data suggest that, on the average, people are able to lose more weight on a low carbohydrate diet than on a low fat diet?
• Response Variable : Weight loss (pounds)
• Explanatory (Grouping) Variable : Type of diet
• Null Hypothesis : There is no difference in the mean amount of weight loss when comparing a low carbohydrate diet with a low fat diet (population mean weight loss on a low carbohydrate diet = population mean weight loss on a low fat diet).
• Alternative Hypothesis : The mean weight loss should be greater for those on a low carbohydrate diet when compared with those on a low fat diet (population mean weight loss on a low carbohydrate diet > population mean weight loss on a low fat diet). This is a one-sided alternative hypothesis.

Example 10.6: Hypotheses about the relationship between Two Categorical Variables Section  

• Research Question : Do the odds of having a stroke increase if you inhale second hand smoke ? A case-control study of non-smoking stroke patients and controls of the same age and occupation are asked if someone in their household smokes.
• Variables : There are two different categorical variables (Stroke patient vs control and whether the subject lives in the same household as a smoker). Living with a smoker (or not) is the natural explanatory variable and having a stroke (or not) is the natural response variable in this situation.
• Null Hypothesis : There is no relationship between whether or not a person has a stroke and whether or not a person lives with a smoker (odds ratio between stroke and second-hand smoke situation is = 1).
• Alternative Hypothesis : There is a relationship between whether or not a person has a stroke and whether or not a person lives with a smoker (odds ratio between stroke and second-hand smoke situation is > 1). This is a one-tailed alternative.

This research question might also be addressed like example 11.4 by making the hypotheses about comparing the proportion of stroke patients that live with smokers to the proportion of controls that live with smokers.

Example 10.7: Hypotheses about the relationship between Two Measurement Variables Section  

• Research Question : A financial analyst believes there might be a positive association between the change in a stock's price and the amount of the stock purchased by non-management employees the previous day (stock trading by management being under "insider-trading" regulatory restrictions).
• Variables : Daily price change information (the response variable) and previous day stock purchases by non-management employees (explanatory variable). These are two different measurement variables.
• Null Hypothesis : The correlation between the daily stock price change (\$) and the daily stock purchases by non-management employees (\$) = 0.
• Alternative Hypothesis : The correlation between the daily stock price change (\$) and the daily stock purchases by non-management employees (\$) > 0. This is a one-sided alternative hypothesis.

Example 10.8: Hypotheses about comparing the relationship between Two Measurement Variables in Two Samples Section  

• Research Question : Is there a linear relationship between the amount of the bill (\$) at a restaurant and the tip (\$) that was left. Is the strength of this association different for family restaurants than for fine dining restaurants?
• Variables : There are two different measurement variables. The size of the tip would depend on the size of the bill so the amount of the bill would be the explanatory variable and the size of the tip would be the response variable.
• Null Hypothesis : The correlation between the amount of the bill (\$) at a restaurant and the tip (\$) that was left is the same at family restaurants as it is at fine dining restaurants.
• Alternative Hypothesis : The correlation between the amount of the bill (\$) at a restaurant and the tip (\$) that was left is the difference at family restaurants then it is at fine dining restaurants. This is a two-sided alternative hypothesis.

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Null hypothesis or what

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Measurements on a (random) sample of babies born to mothers who took “Prescription Drug A” during pregnancy were compared to measurements on a (random) sample of babies born to mothers who did not take “Prescription Drug A”. A statistically significant difference in the mean head circumference was found between children born to the two groups of mothers (p = .03). Based only on this information you can conclude: a. There is a 3% chance the null hypothesis is true. b. Taking “Prescription Drug A” during pregnancy causes a reduction in child’s head circumference. c. Taking “Prescription Drug A” is associated with an increased child head circumference. d. The sample mean difference in children’s head circumference between children born to the two groups of mothers is clinically important/significant. e. None of the above  

a. There is a 3% chance the null hypothesis is true.  

a is the answer.  

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9.1E: Null and Alternative Hypotheses (Exercises)

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Exercise \(\PageIndex{5}\)

You are testing that the mean speed of your cable Internet connection is more than three Megabits per second. What is the random variable? Describe in words.

The random variable is the mean Internet speed in Megabits per second.

Exercise \(\PageIndex{1}\)

You are testing that the mean speed of your cable Internet connection is more than three Megabits per second. State the null and alternative hypotheses.

The American family has an average of two children. What is the random variable? Describe in words.

The random variable is the mean number of children an American family has.

Exercise \(\PageIndex{8}\)

The mean entry level salary of an employee at a company is $58,000. You believe it is higher for IT professionals in the company. State the null and alternative hypotheses.

Exercise \(\PageIndex{9}\)

A sociologist claims the probability that a person picked at random in Times Square in New York City is visiting the area is 0.83. You want to test to see if the proportion is actually less. What is the random variable? Describe in words.

The random variable is the proportion of people picked at random in Times Square visiting the city.

Exercise \(\PageIndex{10}\)

A sociologist claims the probability that a person picked at random in Times Square in New York City is visiting the area is 0.83. You want to test to see if the claim is correct. State the null and alternative hypotheses.

Exercise \(\PageIndex{11}\)

In a population of fish, approximately 42% are female. A test is conducted to see if, in fact, the proportion is less. State the null and alternative hypotheses.

• \(H_{0}: p = 0.42\)
• \(H_{a}: p < 0.42\)

Exercise \(\PageIndex{12}\)

Suppose that a recent article stated that the mean time spent in jail by a first–time convicted burglar is 2.5 years. A study was then done to see if the mean time has increased in the new century. A random sample of 26 first-time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was 3 years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. If you were conducting a hypothesis test to determine if the mean length of jail time has increased, what would the null and alternative hypotheses be? The distribution of the population is normal.

• \(H_{0}\):_______
• \(H_{a}\):_______

Exercise \(\PageIndex{13}\)

A random survey of 75 death row inmates revealed that the mean length of time on death row is 17.4 years with a standard deviation of 6.3 years. If you were conducting a hypothesis test to determine if the population mean time on death row could likely be 15 years, what would the null and alternative hypotheses be?

• \(H_{0}\):_________
• \(H_{a}\):_________
• \(H_{0}: \mu = 15\)
• \(H_{a}: \mu \neq 15\)

The National Institute 9.2.14 of Mental Health published an article stating that in any one-year period, approximately 9.5 percent of American adults suffer from depression or a depressive illness. Suppose that in a survey of 100 people in a certain town, seven of them suffered from depression or a depressive illness. If you were conducting a hypothesis test to determine if the true proportion of people in that town suffering from depression or a depressive illness is lower than the percent in the general adult American population, what would the null and alternative hypotheses be?

Null Hypothesis Examples

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The null hypothesis —which assumes that there is no meaningful relationship between two variables—may be the most valuable hypothesis for the scientific method because it is the easiest to test using a statistical analysis. This means you can support your hypothesis with a high level of confidence. Testing the null hypothesis can tell you whether your results are due to the effect of manipulating ​ the dependent variable or due to chance.

What Is the Null Hypothesis?

The null hypothesis states there is no relationship between the measured phenomenon (the dependent variable) and the independent variable . You do not​ need to believe that the null hypothesis is true to test it. On the contrary, you will likely suspect that there is a relationship between a set of variables. One way to prove that this is the case is to reject the null hypothesis. Rejecting a hypothesis does not mean an experiment was "bad" or that it didn't produce results. In fact, it is often one of the first steps toward further inquiry.

To distinguish it from other hypotheses, the null hypothesis is written as ​ H 0  (which is read as “H-nought,” "H-null," or "H-zero"). A significance test is used to determine the likelihood that the results supporting the null hypothesis are not due to chance. A confidence level of 95 percent or 99 percent is common. Keep in mind, even if the confidence level is high, there is still a small chance the null hypothesis is not true, perhaps because the experimenter did not account for a critical factor or because of chance. This is one reason why it's important to repeat experiments.

Examples of the Null Hypothesis

To write a null hypothesis, first start by asking a question. Rephrase that question in a form that assumes no relationship between the variables. In other words, assume a treatment has no effect. Write your hypothesis in a way that reflects this.

• What Is a Hypothesis? (Science)
• What 'Fail to Reject' Means in a Hypothesis Test
• What Are the Elements of a Good Hypothesis?
• Scientific Method Vocabulary Terms
• Null Hypothesis Definition and Examples
• Definition of a Hypothesis
• Six Steps of the Scientific Method
• Hypothesis Test for the Difference of Two Population Proportions
• Understanding Simple vs Controlled Experiments
• What Is the Difference Between Alpha and P-Values?
• Null Hypothesis and Alternative Hypothesis
• What Are Examples of a Hypothesis?
• What It Means When a Variable Is Spurious
• Hypothesis Test Example
• How to Conduct a Hypothesis Test
• What Is a P-Value?

13.1 Understanding Null Hypothesis Testing

Learning objectives.

• Explain the purpose of null hypothesis testing, including the role of sampling error.
• Describe the basic logic of null hypothesis testing.
• Describe the role of relationship strength and sample size in determining statistical significance and make reasonable judgments about statistical significance based on these two factors.

  The Purpose of Null Hypothesis Testing

As we have seen, psychological research typically involves measuring one or more variables in a sample and computing descriptive statistics for that sample. In general, however, the researcher’s goal is not to draw conclusions about that sample but to draw conclusions about the population that the sample was selected from. Thus researchers must use sample statistics to draw conclusions about the corresponding values in the population. These corresponding values in the population are called  parameters . Imagine, for example, that a researcher measures the number of depressive symptoms exhibited by each of 50 adults with clinical depression and computes the mean number of symptoms. The researcher probably wants to use this sample statistic (the mean number of symptoms for the sample) to draw conclusions about the corresponding population parameter (the mean number of symptoms for adults with clinical depression).

Unfortunately, sample statistics are not perfect estimates of their corresponding population parameters. This is because there is a certain amount of random variability in any statistic from sample to sample. The mean number of depressive symptoms might be 8.73 in one sample of adults with clinical depression, 6.45 in a second sample, and 9.44 in a third—even though these samples are selected randomly from the same population. Similarly, the correlation (Pearson’s  r ) between two variables might be +.24 in one sample, −.04 in a second sample, and +.15 in a third—again, even though these samples are selected randomly from the same population. This random variability in a statistic from sample to sample is called  sampling error . (Note that the term error  here refers to random variability and does not imply that anyone has made a mistake. No one “commits a sampling error.”)

One implication of this is that when there is a statistical relationship in a sample, it is not always clear that there is a statistical relationship in the population. A small difference between two group means in a sample might indicate that there is a small difference between the two group means in the population. But it could also be that there is no difference between the means in the population and that the difference in the sample is just a matter of sampling error. Similarly, a Pearson’s  r  value of −.29 in a sample might mean that there is a negative relationship in the population. But it could also be that there is no relationship in the population and that the relationship in the sample is just a matter of sampling error.

In fact, any statistical relationship in a sample can be interpreted in two ways:

• There is a relationship in the population, and the relationship in the sample reflects this.
• There is no relationship in the population, and the relationship in the sample reflects only sampling error.

The purpose of null hypothesis testing is simply to help researchers decide between these two interpretations.

The Logic of Null Hypothesis Testing

Null hypothesis testing  is a formal approach to deciding between two interpretations of a statistical relationship in a sample. One interpretation is called the  null hypothesis  (often symbolized  H 0  and read as “H-naught”). This is the idea that there is no relationship in the population and that the relationship in the sample reflects only sampling error. Informally, the null hypothesis is that the sample relationship “occurred by chance.” The other interpretation is called the  alternative hypothesis  (often symbolized as  H 1 ). This is the idea that there is a relationship in the population and that the relationship in the sample reflects this relationship in the population.

Again, every statistical relationship in a sample can be interpreted in either of these two ways: It might have occurred by chance, or it might reflect a relationship in the population. So researchers need a way to decide between them. Although there are many specific null hypothesis testing techniques, they are all based on the same general logic. The steps are as follows:

• Assume for the moment that the null hypothesis is true. There is no relationship between the variables in the population.
• Determine how likely the sample relationship would be if the null hypothesis were true.
• If the sample relationship would be extremely unlikely, then reject the null hypothesis  in favor of the alternative hypothesis. If it would not be extremely unlikely, then  retain the null hypothesis .

Following this logic, we can begin to understand why Mehl and his colleagues concluded that there is no difference in talkativeness between women and men in the population. In essence, they asked the following question: “If there were no difference in the population, how likely is it that we would find a small difference of  d  = 0.06 in our sample?” Their answer to this question was that this sample relationship would be fairly likely if the null hypothesis were true. Therefore, they retained the null hypothesis—concluding that there is no evidence of a sex difference in the population. We can also see why Kanner and his colleagues concluded that there is a correlation between hassles and symptoms in the population. They asked, “If the null hypothesis were true, how likely is it that we would find a strong correlation of +.60 in our sample?” Their answer to this question was that this sample relationship would be fairly unlikely if the null hypothesis were true. Therefore, they rejected the null hypothesis in favor of the alternative hypothesis—concluding that there is a positive correlation between these variables in the population.

A crucial step in null hypothesis testing is finding the likelihood of the sample result if the null hypothesis were true. This probability is called the  p value . A low  p  value means that the sample result would be unlikely if the null hypothesis were true and leads to the rejection of the null hypothesis. A p  value that is not low means that the sample result would be likely if the null hypothesis were true and leads to the retention of the null hypothesis. But how low must the  p  value be before the sample result is considered unlikely enough to reject the null hypothesis? In null hypothesis testing, this criterion is called  α (alpha)  and is almost always set to .05. If there is a 5% chance or less of a result as extreme as the sample result if the null hypothesis were true, then the null hypothesis is rejected. When this happens, the result is said to be  statistically significant . If there is greater than a 5% chance of a result as extreme as the sample result when the null hypothesis is true, then the null hypothesis is retained. This does not necessarily mean that the researcher accepts the null hypothesis as true—only that there is not currently enough evidence to reject it. Researchers often use the expression “fail to reject the null hypothesis” rather than “retain the null hypothesis,” but they never use the expression “accept the null hypothesis.”

The Misunderstood  p  Value

The  p  value is one of the most misunderstood quantities in psychological research (Cohen, 1994) [1] . Even professional researchers misinterpret it, and it is not unusual for such misinterpretations to appear in statistics textbooks!

The most common misinterpretation is that the  p  value is the probability that the null hypothesis is true—that the sample result occurred by chance. For example, a misguided researcher might say that because the  p  value is .02, there is only a 2% chance that the result is due to chance and a 98% chance that it reflects a real relationship in the population. But this is incorrect . The  p  value is really the probability of a result at least as extreme as the sample result  if  the null hypothesis  were  true. So a  p  value of .02 means that if the null hypothesis were true, a sample result this extreme would occur only 2% of the time.

You can avoid this misunderstanding by remembering that the  p  value is not the probability that any particular  hypothesis  is true or false. Instead, it is the probability of obtaining the  sample result  if the null hypothesis were true.

“Null Hypothesis” retrieved from http://imgs.xkcd.com/comics/null_hypothesis.png (CC-BY-NC 2.5)

Role of Sample Size and Relationship Strength

Recall that null hypothesis testing involves answering the question, “If the null hypothesis were true, what is the probability of a sample result as extreme as this one?” In other words, “What is the  p  value?” It can be helpful to see that the answer to this question depends on just two considerations: the strength of the relationship and the size of the sample. Specifically, the stronger the sample relationship and the larger the sample, the less likely the result would be if the null hypothesis were true. That is, the lower the  p  value. This should make sense. Imagine a study in which a sample of 500 women is compared with a sample of 500 men in terms of some psychological characteristic, and Cohen’s  d  is a strong 0.50. If there were really no sex difference in the population, then a result this strong based on such a large sample should seem highly unlikely. Now imagine a similar study in which a sample of three women is compared with a sample of three men, and Cohen’s  d  is a weak 0.10. If there were no sex difference in the population, then a relationship this weak based on such a small sample should seem likely. And this is precisely why the null hypothesis would be rejected in the first example and retained in the second.

Of course, sometimes the result can be weak and the sample large, or the result can be strong and the sample small. In these cases, the two considerations trade off against each other so that a weak result can be statistically significant if the sample is large enough and a strong relationship can be statistically significant even if the sample is small. Table 13.1 shows roughly how relationship strength and sample size combine to determine whether a sample result is statistically significant. The columns of the table represent the three levels of relationship strength: weak, medium, and strong. The rows represent four sample sizes that can be considered small, medium, large, and extra large in the context of psychological research. Thus each cell in the table represents a combination of relationship strength and sample size. If a cell contains the word  Yes , then this combination would be statistically significant for both Cohen’s  d  and Pearson’s  r . If it contains the word  No , then it would not be statistically significant for either. There is one cell where the decision for  d  and  r  would be different and another where it might be different depending on some additional considerations, which are discussed in Section 13.2 “Some Basic Null Hypothesis Tests”

Although Table 13.1 provides only a rough guideline, it shows very clearly that weak relationships based on medium or small samples are never statistically significant and that strong relationships based on medium or larger samples are always statistically significant. If you keep this lesson in mind, you will often know whether a result is statistically significant based on the descriptive statistics alone. It is extremely useful to be able to develop this kind of intuitive judgment. One reason is that it allows you to develop expectations about how your formal null hypothesis tests are going to come out, which in turn allows you to detect problems in your analyses. For example, if your sample relationship is strong and your sample is medium, then you would expect to reject the null hypothesis. If for some reason your formal null hypothesis test indicates otherwise, then you need to double-check your computations and interpretations. A second reason is that the ability to make this kind of intuitive judgment is an indication that you understand the basic logic of this approach in addition to being able to do the computations.

Statistical Significance Versus Practical Significance

Table 13.1 illustrates another extremely important point. A statistically significant result is not necessarily a strong one. Even a very weak result can be statistically significant if it is based on a large enough sample. This is closely related to Janet Shibley Hyde’s argument about sex differences (Hyde, 2007) [2] . The differences between women and men in mathematical problem solving and leadership ability are statistically significant. But the word  significant  can cause people to interpret these differences as strong and important—perhaps even important enough to influence the college courses they take or even who they vote for. As we have seen, however, these statistically significant differences are actually quite weak—perhaps even “trivial.”

This is why it is important to distinguish between the  statistical  significance of a result and the  practical  significance of that result.  Practical significance refers to the importance or usefulness of the result in some real-world context. Many sex differences are statistically significant—and may even be interesting for purely scientific reasons—but they are not practically significant. In clinical practice, this same concept is often referred to as “clinical significance.” For example, a study on a new treatment for social phobia might show that it produces a statistically significant positive effect. Yet this effect still might not be strong enough to justify the time, effort, and other costs of putting it into practice—especially if easier and cheaper treatments that work almost as well already exist. Although statistically significant, this result would be said to lack practical or clinical significance.

“Conditional Risk” retrieved from http://imgs.xkcd.com/comics/conditional_risk.png (CC-BY-NC 2.5)

Key Takeaways

• Null hypothesis testing is a formal approach to deciding whether a statistical relationship in a sample reflects a real relationship in the population or is just due to chance.
• The logic of null hypothesis testing involves assuming that the null hypothesis is true, finding how likely the sample result would be if this assumption were correct, and then making a decision. If the sample result would be unlikely if the null hypothesis were true, then it is rejected in favor of the alternative hypothesis. If it would not be unlikely, then the null hypothesis is retained.
• The probability of obtaining the sample result if the null hypothesis were true (the  p  value) is based on two considerations: relationship strength and sample size. Reasonable judgments about whether a sample relationship is statistically significant can often be made by quickly considering these two factors.
• Statistical significance is not the same as relationship strength or importance. Even weak relationships can be statistically significant if the sample size is large enough. It is important to consider relationship strength and the practical significance of a result in addition to its statistical significance.
• Discussion: Imagine a study showing that people who eat more broccoli tend to be happier. Explain for someone who knows nothing about statistics why the researchers would conduct a null hypothesis test.
• The correlation between two variables is  r  = −.78 based on a sample size of 137.
• The mean score on a psychological characteristic for women is 25 ( SD  = 5) and the mean score for men is 24 ( SD  = 5). There were 12 women and 10 men in this study.
• In a memory experiment, the mean number of items recalled by the 40 participants in Condition A was 0.50 standard deviations greater than the mean number recalled by the 40 participants in Condition B.
• In another memory experiment, the mean scores for participants in Condition A and Condition B came out exactly the same!
• A student finds a correlation of  r  = .04 between the number of units the students in his research methods class are taking and the students’ level of stress.
• Cohen, J. (1994). The world is round: p < .05. American Psychologist, 49 , 997–1003. ↵
• Hyde, J. S. (2007). New directions in the study of gender similarities and differences. Current Directions in Psychological Science, 16 , 259–263. ↵

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• v.29(6); 2017 Dec 25

Language: English | Chinese

Comparisons of Superiority, Non-inferiority, and Equivalence Trials

有效性试验、非劣效性试验、和等效试验之间的比较.

1 Department of Biostatistics and Computational Biology, University of Rochester, Rochester, NY, USA

Hongyue WANG

3 Division of Biostatistics, University of California San Diego, La Jolla, CA, USA

Changyong FENG

2 Department of Anesthesiology, University of Rochester, Rochester, NY, USA

Efficacy of a new drug or treatment is usually established through randomized clinical trials. However, specifying hypotheses remains a challenging problem for biomedical researchers. In this survey we discuss superiority, non-inferiority, and equivalence trials. These three types of trials have different assumptions on treatment effects. We compare the assumptions underlying these trials and provide sample size formulas.

概述

一种新型药物或治疗的效果通常是通过临床随机对照试验获得的。然而，假设的设定对生物医学研究者来说仍然是具有挑战性的问题。在本调查中，我们讨论了有效性试验、非劣效性试验、和等效性试验。这三类试验对治疗效果均有不同的假设。我们比较了这些试验中的假设并提供了样本量计算公式。

1. Introduction

In medical research, randomized clinical trials are the gold standard for establishing efficacy of a newly developed drug/treatment method. [ 1-5 ] A well designed clinical trial should clearly specify the kind of hypothesis to be tested and procedures to be used for analysis of primary outcomes. For example, depending on the purpose of the trial, we need to specify whether the study is to test superiority (i.e., better than), non-inferiority, or equivalence, between different treatment conditions. Sample size calculation, data analysis, and interpretation of analysis results all depend on the type of hypothesis specified. From our interactions with biomedical and psychosocial researchers, these issues do not seem to be clear and appreciated in the research community. In this report, we attempt to clarify different types of hypothesis testing and rationales for each, and show how to calculate sample size in each case.

Hypotheses in most clinical trials can be stated in terms of differences in the mean response of an outcome of interest such as group means. For example, prostate-specific antigen (PSA) level is a common outcome for prostate cancer patients [ 6 ] (or the Hamilton Depression Rating Scale (HAM-D) is a popular scale for depression severity). In this case, PSA level is a continuous measure and the hypothesis is stated to compare mean PSA levels between two groups. Sometimes, an outcome of interest may be categorical. For example, the outcome may be the survival status of the patient by the end of the follow-up (or diagnosis of clinical depression). In this case, for each patient we can use a binary outcome variable X with value 1 (0) to denote the survival (death) of the patient. The proportion of survival in each group is just the mean value of X for patients in the corresponding group. The hypothesis to compare differences in prevalence of depression between two study populations can be stated in terms of difference between the means of X for the two groups.

We think technical difficulty may likely be responsible for the confusion. Thus, we will try to make our presentation as non-technical as possible. Also, for simplicity we assume two groups with i = 0, 1, denoting the control and treatment groups. For group i, let X ii denote the primary outcome of the jth subject in the ith group. Let µ i and σ i 2 denote the mean and variance of X ij in group i. They are also loosely called the group mean and variance. We further assume higher value mean of X ij means better outcome. Hence the treatment group is said to be ‘better’ than the control group if µ 1 >µ 0 .

In the following sections, we introduce the three types of trials: superiority, non-inferiority, and equivalence trial. We start with the most popular superiority.

2. Superiority trial

In a superiority trial, we want to show that the new treatment intervention (drug, psychotherapy) is superior to (better than) the control condition. For example, we want to know if a new drug can significantly increase CD4 counts for HIV patients or a novel psychosocial therapy will increase social activities for lonely old adults.

For many researchers, a challenging problem is how to specify the null and alternative hypotheses for the specific trial. A rule of thumb is to specify the null hypothesis opposite to what we expect for the outcome. For example, if we want to test if treatment A is better than treatment B , the null hypothesis is that A is not better than or same as B . We anticipate that the data from the trial will tell us otherwise and reject the null hypothesis in support of the anticipated superiority of treatment A . Based on this idea, the null and alternative hypotheses of a superiority trial are specified as

where δ ≥0.

Under the null hypothesis, the mean value of the treatment group is less than or equal to that of the control group plus a nonnegative number δ . Sometimes we may not feel so confident that the treatment is better than the control, even if the mean value of the treatment group is really greater than the control, but the difference is small. For example, suppose that we want to test if a new instructional method improves the performance of students in a math test. If the new method increases the average score from 75 to 76, we may be reluctant to say that the new method is better than the current one. However, if the new method can increase the average score by at least 6 points, then we may think that the new method is superior to the current one. These 6 points is the superiority margin of the new instructional method. If the improvement of the new method is less than this value, we may not care much about it even if it has a higher group mean.

The value δ in (1) is called margin of clinical significance. [ 4 ] For a given study, the larger the δ , the harder to reject the null hypothesis, as reflected in the sample size formula in (2) and (3) below. Therefore, this margin is the threshold for which we claim the superiority of the new treatment. For different studies, choices of δ depend on the contexts of the study and scale of measures. For example, for a study on suicide rate, even a small δ in reducing the rate of suicide will have a significant impact on the lives of those at risk for suicide. For a new method to improve scores on a math test, a difference of 6 points or higher may be a reasonable threshold for adopting the new method. There is no general rule to specify the margin. It depends on the purpose of the study.

In most studies, different groups typically have equal sample sizes. However, sometimes we may want assign more subjects to one or more groups. For example, for a study with two active treatments and one control, we may want to have a larger sample size for the control for more power to compare the treatment with the control. Below, we consider a general situation and provide formulas of sample size calculations for unequal sample sizes for the two groups. We assume that n 1 = rn 0 , where r is a fixed positive constant.

For a constant η ∈ (0,1), let z n denote the ηth upper-quantile of standard normal distribution. For example, if η = 0.1, then z n = 1.2816, which means that for a random variable with standard normal distribution, it is greater than 1.2816 with 10% probability. For each real number x, let [x] denote the least integer greater than or equal to x. For example, [ 8 ]=8 and [8.1]=9.

Sample size depends on the true mean difference, d, standard deviations for the two groups, and a level of significance α (type | error), and the power. Given all these parameters, required sample sizes for the treatment and control groups are as follows:

where β = 1– power and is called the type II error. For example, if power = 80%, then β = 0.2. The total sample size n=n 1 +n 0 is minimized when r = σ 1 /σ 0 . In most studies, we assume equal group variance, i.e., σ 1 = σ 0 , the minimum overall sample achieves when r = 1. This fact may explain why most clinical trials use equal sample sizes in two groups.

Given d, sample sizes increase with δ . Therefore, it becomes more difficult to reject the null hypothesis if the margin of clinical significance δ is set higher.

Remark. Although the hypotheses in (1) are very natural and intuitive for the superiority trial, there are many discussions about the establishment of superiority from regulatory agencies, see for example Dunnett and Gent [ 3 ] , Lesaffre [ 7 ] , Sackett [ 8 ] , and Sackett. [ 9 ] According to Chow and Liu, [ 1 ] testing of superiority is usually done in two steps. The first step is to show the treatment and groups are significantly different by testing the hypotheses

If the null hypothesis in (4) is rejected, then check if the sample mean value in the treatment group is larger than the control. If it is, then we claim that the treatment group is superior to the control. According to Chow and Liu [ 1 ] , this two-step procedure is equivalent to testing the superiority based on the following special form of (1)

with significance level α/2.

3. Non-inferiority trial

A non-inferiority trial is to show that treatment A is not worse than the treatment B. Although these kinds of trials are not used to establish better treatment efficacy, the new method may have advantages over current methods in other aspects. For example, the new intervention may be less costly, less invasive, and have less side effects.

The hypotheses of non-inferiority clinical trials are

where δ ≥ 0 and is also called the margin of clinical significance which is usually small.

The non-inferiority of the treatment to the control can be easily understood form the alternative hypothesis. If the mean difference between the treatment and control group is greater than δ , then the treatment is non-inferior to the control. Unlike the superiority trial, we don’t need the treatment to be better than the control. For example, if δ > 0, the treatment may be ‘worse’ than the control (i.e. µ 1 – µ 0 < 0). However, as long as µ 1 – µ 0 > - δ , the treatment is the non-inferior.

By comparing (1) and (5), we may see that it is generally easier to establish the non-inferiority than superiority. This is true if we compare the sample size formulas in these two cases. Suppose the true mean difference µ 1 – µ 0 is d. Given significance level a and power 1-β, the required sample sizes in the treatment and control groups in a non-inferiority trial are

It’s easy to see that given d, n 0 increases with δ. This is very intuitive. The larger the δ, the easier to reject the null hypothesis.

4. Equivalence trial

‘Equivalence’ does not mean ‘equal’ or ’same’ as in practice. When we say the treatment and the control are equivalent, we mean that they are ‘similar’. By quantifying ‘Similarity’ using a tolerance range, the hypotheses for an equivalence trial are specified as

where δ > 0 is a pre-specified tolerance margin. If the null hypothesis is rejected, then the mean difference of two groups is within the tolerance range and the treatment and control are equivalent.

A closer look at (6) shows the hypotheses in an equivalence trial are the same as

Comparing (5) and (6) we can see that the equivalence trial is the intersection of two non-inferiority trials. Intuitively, the treatment and control are equivalent, if and only if neither one is inferior to the other.

Suppose the true mean difference µ 1 – µ 0 is d. Given significance level and power 1-β, the required sample sizes in the treatment and control groups in an equivalence trial are

It’s easy to see that given d, n 0 increases with δ. Thus, the larger the δ, the easier to reject the null hypothesis.

5. Conclusion

Superiority, non-inferiority, and equivalence trials are three types of widely used clinical trials. By a close examination of these hypotheses we can see that there are some similarities between trials. For example, superiority is a special case of non-inferiority. It is much easier to establish non-inferiority than superiority. Equivalence is the combination of two non-inferiority trials. On the analytic side, as different types of trials entail quite different interpretations and sample sizes, we must pay close attention to their different uses and use the right type of study in a given situation.

Bokai Wang obtained his BS in Statistics from the Nankai University in 2010 and his MS in Applied Statistics from the Bowling Green State University (Bowling Green, OH) in 2012. He is currently a PhD student in Statistics at the University of Rochester. His research interests include Survival Analysis, Causal Inference, and Variable Selection. Currently, he has published 6 papers in peer reviewed journals.

Funding statement

This study received no external funding.

Conflicts of interest statement

The authors have no conflict of interest to declare.

Authors’ contributions

Bokai Wang, Hongyue Wang: Theoretical derivation and manuscript drafting.

Xin M. Tu, Changyong Feng: Manuscript editing.

• Math Article

Null Hypothesis

In mathematics, Statistics deals with the study of research and surveys on the numerical data. For taking surveys, we have to define the hypothesis. Generally, there are two types of hypothesis. One is a null hypothesis, and another is an alternative hypothesis .

In probability and statistics, the null hypothesis is a comprehensive statement or default status that there is zero happening or nothing happening. For example, there is no connection among groups or no association between two measured events. It is generally assumed here that the hypothesis is true until any other proof has been brought into the light to deny the hypothesis. Let us learn more here with definition, symbol, principle, types and example, in this article.

Table of contents:

• Comparison with Alternative Hypothesis

Null Hypothesis Definition

The null hypothesis is a kind of hypothesis which explains the population parameter whose purpose is to test the validity of the given experimental data. This hypothesis is either rejected or not rejected based on the viability of the given population or sample . In other words, the null hypothesis is a hypothesis in which the sample observations results from the chance. It is said to be a statement in which the surveyors wants to examine the data. It is denoted by H 0 .

Null Hypothesis Symbol

In statistics, the null hypothesis is usually denoted by letter H with subscript ‘0’ (zero), such that H 0 . It is pronounced as H-null or H-zero or H-nought. At the same time, the alternative hypothesis expresses the observations determined by the non-random cause. It is represented by H 1 or H a .

Null Hypothesis Principle

The principle followed for null hypothesis testing is, collecting the data and determining the chances of a given set of data during the study on some random sample, assuming that the null hypothesis is true. In case if the given data does not face the expected null hypothesis, then the outcome will be quite weaker, and they conclude by saying that the given set of data does not provide strong evidence against the null hypothesis because of insufficient evidence. Finally, the researchers tend to reject that.

Null Hypothesis Formula

Here, the hypothesis test formulas are given below for reference.

The formula for the null hypothesis is:

H 0 :  p = p 0

The formula for the alternative hypothesis is:

H a = p >p 0 , < p 0 ≠ p 0

The formula for the test static is:

Remember that,  p 0  is the null hypothesis and p – hat is the sample proportion.

Also, read:

Types of Null Hypothesis

There are different types of hypothesis. They are:

Simple Hypothesis

It completely specifies the population distribution. In this method, the sampling distribution is the function of the sample size.

Composite Hypothesis

The composite hypothesis is one that does not completely specify the population distribution.

Exact Hypothesis

Exact hypothesis defines the exact value of the parameter. For example μ= 50

Inexact Hypothesis

This type of hypothesis does not define the exact value of the parameter. But it denotes a specific range or interval. For example 45< μ <60

Null Hypothesis Rejection

Sometimes the null hypothesis is rejected too. If this hypothesis is rejected means, that research could be invalid. Many researchers will neglect this hypothesis as it is merely opposite to the alternate hypothesis. It is a better practice to create a hypothesis and test it. The goal of researchers is not to reject the hypothesis. But it is evident that a perfect statistical model is always associated with the failure to reject the null hypothesis.

How do you Find the Null Hypothesis?

The null hypothesis says there is no correlation between the measured event (the dependent variable) and the independent variable. We don’t have to believe that the null hypothesis is true to test it. On the contrast, you will possibly assume that there is a connection between a set of variables ( dependent and independent).

When is Null Hypothesis Rejected?

The null hypothesis is rejected using the P-value approach. If the P-value is less than or equal to the α, there should be a rejection of the null hypothesis in favour of the alternate hypothesis. In case, if P-value is greater than α, the null hypothesis is not rejected.

Null Hypothesis and Alternative Hypothesis

Now, let us discuss the difference between the null hypothesis and the alternative hypothesis.

Null Hypothesis Examples

Here, some of the examples of the null hypothesis are given below. Go through the below ones to understand the concept of the null hypothesis in a better way.

If a medicine reduces the risk of cardiac stroke, then the null hypothesis should be “the medicine does not reduce the chance of cardiac stroke”. This testing can be performed by the administration of a drug to a certain group of people in a controlled way. If the survey shows that there is a significant change in the people, then the hypothesis is rejected.

Few more examples are:

1). Are there is 100% chance of getting affected by dengue?

Ans: There could be chances of getting affected by dengue but not 100%.

2). Do teenagers are using mobile phones more than grown-ups to access the internet?

Ans: Age has no limit on using mobile phones to access the internet.

3). Does having apple daily will not cause fever?

Ans: Having apple daily does not assure of not having fever, but increases the immunity to fight against such diseases.

4). Do the children more good in doing mathematical calculations than grown-ups?

Ans: Age has no effect on Mathematical skills.

In many common applications, the choice of the null hypothesis is not automated, but the testing and calculations may be automated. Also, the choice of the null hypothesis is completely based on previous experiences and inconsistent advice. The choice can be more complicated and based on the variety of applications and the diversity of the objectives. 

The main limitation for the choice of the null hypothesis is that the hypothesis suggested by the data is based on the reasoning which proves nothing. It means that if some hypothesis provides a summary of the data set, then there would be no value in the testing of the hypothesis on the particular set of data. 

Frequently Asked Questions on Null Hypothesis

What is meant by the null hypothesis.

In Statistics, a null hypothesis is a type of hypothesis which explains the population parameter whose purpose is to test the validity of the given experimental data.

What are the benefits of hypothesis testing?

Hypothesis testing is defined as a form of inferential statistics, which allows making conclusions from the entire population based on the sample representative.

When a null hypothesis is accepted and rejected?

The null hypothesis is either accepted or rejected in terms of the given data. If P-value is less than α, then the null hypothesis is rejected in favor of the alternative hypothesis, and if the P-value is greater than α, then the null hypothesis is accepted in favor of the alternative hypothesis.

Why is the null hypothesis important?

The importance of the null hypothesis is that it provides an approximate description of the phenomena of the given data. It allows the investigators to directly test the relational statement in a research study.

How to accept or reject the null hypothesis in the chi-square test?

If the result of the chi-square test is bigger than the critical value in the table, then the data does not fit the model, which represents the rejection of the null hypothesis.

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