statistics assignment 4.20

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4.20: Assignment- Scatterplot

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• Page ID 251335

In this exercise we will:

• Learn how to create a scatterplot.
• Use the scatterplot to examine the relationship between two quantitative variables.
• Learn how to create a labeled scatterplot.
• Use the labeled scatterplot to better understand the form of a relationship.

In this activity we explore the relationship between weight and height for 81 adults. We will use height as the explanatory variable. Weight is the response variable.

We will then label the men and women by adding the categorical variable gender to the scatterplot. We will see if separating the groups contributes to our understanding of the form of the relationship between height and weight.

Instructions

Click on the link corresponding to your statistical package to see instructions for completing the activity, and then answer the questions below.

R | StatCrunch | Minitab | Excel 2007 | TI Calculator

Question 1:

Describe the relationship between the height and weight of the subjects. To describe the relationship write about the pattern (direction, form, and strength) and any deviations from the pattern (outliers).

So far we have studied the relationship between height and weight for all of the males and females together. It may be interesting to examine whether the relationship between height and weight is different for males and females. To visualize the effect of the third variable, gender, we will indicate in the scatterplot which observations are males and which are females.

Question 2:

Compare and contrast the relationship between height and weight for males and females. To compare and contrast the relationships by gender write about the pattern (direction, form, and strength) and any deviations from the pattern (outliers) for each group.

Discuss how the patterns for the two groups are similar and how they are different.

• Concepts in Statistics. Provided by : Open Learning Initiative. Located at : http://oli.cmu.edu . License : CC BY: Attribution

0.10 + 0.05 = 0.15

0.35 + 0.40 + 0.10 = 0.85

1(0.15) + 2(0.35) + 3(0.40) + 4(0.10) = 0.15 + 0.70 + 1.20 + 0.40 = 2.45

Let X = the number of events Javier volunteers for each month.

1 – 0.05 = 0.95

0.2 + 1.2 + 2.4 + 1.6 = 5.4

The values of P ( x ) do not sum to one.

Let X = the number of years a physics major will spend doing post-graduate research.

1 – 0.35 – 0.20 – 0.15 – 0.10 – 0.05 = 0.15

1(0.35) + 2(0.20) + 3(0.15) + 4(0.15) + 5(0.10) + 6(0.05) = 0.35 + 0.40 + 0.45 + 0.60 + 0.50 + 0.30 = 2.6 years

X is the number of years a student studies ballet with the teacher.

0.10 + 0.05 + 0.10 = 0.25

The sum of the probabilities sum to one because it is a probability distribution.

− 2 ( 40 52 ) + 30 ( 12 52 ) = − 1.54 + 6.92 = 5.38 − 2 ( 40 52 ) + 30 ( 12 52 ) = − 1.54 + 6.92 = 5.38

X = the number that reply “yes”

0, 1, 2, 3, 4, 5, 6, 7, 8

X = the number of freshmen selected from the study until one replied "yes" that same-sex couples should have the right to legal marital status.

X = the number of business majors in the sample.

2, 3, 4, 5, 6, 7, 8, 9

0, 1, 2, 3, 4, …

X = the number of U.S. teens who die from motor vehicle injuries per day.

0, 1, 2, 3, 4, ...

The variable of interest is X , or the gain or loss, in dollars.

The face cards jack, queen, and king. There are (3)(4) = 12 face cards and 52 – 12 = 40 cards that are not face cards.

We first need to construct the probability distribution for X . We use the card and coin events to determine the probability for each outcome, but we use the monetary value of X to determine the expected value.

• Expected value = ( 6 ) ( 6 52 ) + ( 2 ) ( 6 52 ) + ( − 2 ) ( 40 52 ) = – 32 52 Expected value = ( 6 ) ( 6 52 ) + ( 2 ) ( 6 52 ) + ( − 2 ) ( 40 52 ) = – 32 52
• Expected value = –$0.62, rounded to the nearest cent
• If you play this game repeatedly, over a long string of games, you would expect to lose 62 cents per game, on average.
• You should not play this game to win money because the expected value indicates an expected average loss.
• $200,000; $600,000; $400,000
• third investment because it has the lowest probability of loss
• first investment because it has the highest probability of loss
• second investment

Let X = the amount of money to be won on a ticket. The following table shows the PDF for X .

Calculate the expected value of X .

0(0.969) + 5(0.025) + 25(0.005) + 100(0.001) = 0.35

A fair price for a ticket is $0.35. Any price over $0.35 will enable the lottery to raise money.

X = the number of patients calling in claiming to have the flu, who actually have the flu.

X = 0, 1, 2, ...25

• X = the number of DVDs a Video to Go customer rents
• X = number of questions answered correctly
• X ~ B ( 32,  1 3 ) ( 32,  1 3 )
• We are interested in MORE THAN 75% of 32 questions correct. 75% of 32 is 24. We want to find P ( x > 24). The event "more than 24" is the complement of "less than or equal to 24."
• Using your calculator's distribution menu: 1 – binomcdf ( 32,  1 3 ,  24 ) ( 32,  1 3 ,  24 )
• P ( x > 24) = 0
• The probability of getting more than 75% of the 32 questions correct when randomly guessing is very small and practically zero.
• X = the number of college and universities that offer online offerings.
• 0, 1, 2, …, 13
• X ~ B (13, 0.96)
• P ( x = 12) = 0.3186 P ( x = 13) = 0.5882 More likely to get 13.
• X = the number of fencers who do not use the foil as their main weapon
• 0, 1, 2, 3,... 25
• X ~ B (25,0.40)
• The probability that all 25 not use the foil is almost zero. Therefore, it would be very surprising.
• X = the number of audits in a 20-year period
• 0, 1, 2, …, 20
• X ~ B (20, 0.02)
• X = the number of matches
• X ~ B ( 3 , 1 6 ) ( 3 , 1 6 )
• In dollars: −1, 1, 2, 3
• Multiply each Y value by the corresponding X probability from the PDF table. The answer is −0.0787. You lose about eight cents, on average, per game.
• The house has the advantage.
• Mean = μ = np = 15(0.281) = 4.215
• Standard Deviation = σ = n p q n p q = 15 ( 0.281 ) ( 0.719 ) 15 ( 0.281 ) ( 0.719 ) = 1.7409
• P ( x > 5) = 1 – P ( x ≤ 5) = 1 – binomcdf(15, 0.281, 5) = 1 – 0.7754 = 0.2246 P ( x = 3) = binompdf(15, 0.281, 3) = 0.1927 P ( x = 4) = binompdf(15, 0.281, 4) = 0.2259 It is more likely that four people are literate that three people are.
• X = the number of adults in America who are surveyed until one says he or she will watch the Super Bowl.
• X ~ G (0.40)
• X = the number of pages that advertise footwear
• X takes on the values 0, 1, 2, ..., 20
• X ~ B (20, 29 192 29 192 )
• X = the number of pages we must survey until we find one that advertises footwear. X ~ G ( 29 192 29 192 )
• 6.6207 pages

0, 1, 2, and 3

• X ~ G (0.25)
• Mean = μ = 1 p 1 p = 1 0.25 1 0.25 = 4
• Standard Deviation = σ = 1 − p p 2 1 − p p 2 = 1 − 0 .25 0.25 2 1 − 0 .25 0.25 2 ≈ 3.4641
• P ( x = 10) = geometpdf(0.25, 10) = 0.0188
• P ( x = 20) = geometpdf(0.25, 20) = 0.0011
• P ( x ≤ 5) = geometcdf(0.25, 5) = 0.7627
• 0, 1, 2, 3, ..., 20
• X ~ H (29, 163, 20); r = 29, b = 163, n = 20
• X = the number of Patriots picked
• 0, 1, 2, 3, 4
• X ~ H (4, 8, 9)
• Without replacement
• X ~ P (5.5); μ = 5.5; σ  =  5.5 σ  =  5.5 ≈ 2.3452
• P ( x ≤ 6) = poissoncdf(5.5, 6) ≈ 0.6860
• There is a 15.7% probability that the law staff will receive more calls than they can handle.
• P ( x > 8) = 1 – P ( x ≤ 8) = 1 – poissoncdf(5.5, 8) ≈ 1 – 0.8944 = 0.1056

Let X = the number of defective bulbs in a string.

Using the Poisson distribution:

• μ = np = 100(0.03) = 3
• P ( x ≤ 4) = poissoncdf(3, 4) ≈ 0.8153

Using the binomial distribution:

• X ~ B (100, 0.03)
• P ( x ≤ 4) = binomcdf(100, 0.03, 4) ≈ 0.8179

The Poisson approximation is very good—the difference between the probabilities is only 0.0026.

• X = the number of children for a Spanish woman
• 0, 1, 2, 3,...
• X ~ P (1.47)
• X = the number of fortune cookies that have an extra fortune
• 0, 1, 2, 3,... 144
• X ~ B (144, 0.03) or P (4.32)
• 0.0124 or 0.0133
• 0.6300 or 0.6264
• As n gets larger, the probabilities get closer together.
• X = the number of people audited in one year
• 0, 1, 2, ..., 100
• X = the number of shell pieces in one cake
• X ~ P (1.5)

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• Advanced Sampling Techniques: A Student's Guide to Precision

Advanced Sampling Techniques: Ensuring Accuracy in Survey-Based Research

In contemporary academic research, survey-based studies serve as a prevalent method for data collection and insight generation. The crux of their effectiveness lies in the meticulous application of sampling techniques, a factor that fundamentally shapes the reliability and validity of research findings. This blog serves as an in-depth exploration of advanced sampling techniques, offering a thorough guide tailored to students aiming to refine the precision of their survey-based research. By comprehending and adeptly applying these advanced methods, students can markedly elevate the caliber of their assignments and research undertakings. The nuanced understanding and proficient application of these techniques not only fortify the credibility of findings but also contribute significantly to the broader landscape of scientific inquiry, ensuring that knowledge derived from surveys stands on a robust foundation of accuracy and validity. If you need help with your statistics assignment , I'm here to provide expert assistance and support to ensure your success in mastering advanced sampling techniques and statistical analysis.

Importance of Sampling in Research

Sampling is the bedrock of any research endeavor, forming the crucial link between a study's design and its real-world applicability. The methodological decisions made in the sampling process significantly shape the outcomes of research studies. It's not merely about selecting a subset of individuals or elements from a larger population; rather, it's about crafting a representative microcosm that mirrors the broader context accurately.

The implications of sampling stretch beyond statistical considerations; they extend to the very essence of the research process. The choice of sampling techniques directly influences the generalizability of findings, the robustness of statistical inferences, and the overall validity of research outcomes. For students, recognizing the pivotal role of sampling in research is fundamental. It's not just a technical aspect but a strategic decision that can either fortify or compromise the entire research endeavor. Understanding the importance of sampling equips students with the knowledge needed to navigate the complexities of survey-based studies, enhancing their ability to produce rigorous and insightful academic work.

The Foundation of Research

Sampling serves as the foundational pillar in any research endeavor, a critical step in the pursuit of meaningful insights. At its core, it involves the meticulous selection of a subset of individuals or elements from a larger population, with the ultimate goal of accurately representing the entirety. This process enables researchers to make informed inferences about the broader population, leveraging the efficiency of a more manageable sample size. The robustness of any research study relies heavily on the soundness of its sampling methodology.

Implications for Assignments

Within the realm of academic assignments, the ramifications of choosing appropriate sampling techniques reverberate throughout the entire research process. The selection of an inadequate or biased sample can introduce distortions and errors, ultimately leading to flawed conclusions. The repercussions of such missteps extend beyond the confines of the assignment, significantly impacting the overall quality of the research. Therefore, it is imperative for students to delve deeply into the intricacies of advanced sampling methods, recognizing their role in not just completing assignments but in elevating the credibility and reliability of their academic work.

Common Sampling Techniques

Sampling techniques play a pivotal role in the methodology of survey-based research, and a solid understanding of common methods is essential for researchers and students alike.

1: Simple Random Sampling

Simple random sampling is a fundamental technique that provides each member of the population an equal chance of being selected, forming the basis for more complex methods. Its simplicity facilitates the comprehension of sampling processes, making it an ideal starting point for students.

2: Stratified Sampling

Stratified sampling involves dividing the population into distinct strata based on relevant characteristics before random sampling within each stratum. This method ensures representation from various subgroups, offering a nuanced understanding of diverse perspectives within the larger population.

3: Cluster Sampling

Cluster sampling simplifies the sampling process by grouping the population into clusters and randomly selecting entire clusters for the study. This approach is efficient when a geographically dispersed population poses challenges in direct individual sampling.

4: Systematic Sampling

Systematic sampling is a methodical approach that involves selecting every kth individual from a list after randomly choosing the starting point. Its systematic nature balances simplicity with efficiency, making it suitable for large populations.

A grasp of these common sampling techniques is fundamental for students venturing into survey-based research, providing a solid foundation for more advanced methodologies in their academic journey.

Advanced Sampling Techniques

Navigating the landscape of advanced sampling techniques is a pivotal aspect of conducting rigorous survey-based research. As researchers delve beyond the basics, they encounter a spectrum of methodologies that demand a nuanced understanding. These advanced techniques are instrumental in overcoming specific challenges inherent in diverse research contexts.

Understanding and implementing advanced sampling methods empower researchers to tailor their approaches to the intricacies of their study, ensuring a more precise and nuanced representation of the population of interest. From the strategic selection of participants in purposive sampling to the dynamic nature of snowball sampling, each technique brings its own set of advantages and considerations.

As students venture into the realm of advanced sampling, they gain a broader toolkit to address the complexities of real-world scenarios. This knowledge not only enriches their academic endeavors but also equips them to contribute meaningfully to the evolving landscape of survey-based research. The exploration of advanced sampling techniques marks a significant step forward in the pursuit of accurate, reliable, and impactful research outcomes.

1. Snowball Sampling

Snowball sampling, a non-random technique, relies on existing participants recruiting future participants. This approach proves particularly beneficial when the target population is challenging to reach or identify. While not universally applicable, students should comprehend its application and limitations in specific research contexts. The method fosters a network effect, where the sample size grows organically through participant referrals, offering insights into hidden or marginalized populations. Researchers employing snowball sampling must navigate potential biases introduced by the non-random nature of participant recruitment, emphasizing transparency in reporting methodologies.

2. Quota Sampling

Quota sampling strategically establishes specific quotas for demographic groups, facilitating a non-random but representative sample. This method becomes invaluable when researchers aim to ensure a proportional representation of various subgroups within the population. Students should delve into the nuances of quota sampling, recognizing its potential biases and the importance of defining quotas based on relevant demographic variables. By grasping the intricacies of quota sampling, students can employ this method effectively to enhance the external validity of their research.

3. Convenience Sampling

Convenience sampling, while expedient, involves selecting individuals readily available and willing to participate, potentially introducing biases. Students need to discern the limitations of convenience sampling and its appropriateness in specific research situations. This method is often employed in exploratory research or when access to the entire population is impractical. Students must recognize the trade-offs between convenience and representativeness, emphasizing transparency in acknowledging the potential limitations and biases inherent in this approach.

4. Purposive Sampling

Purposive sampling entails the deliberate selection of individuals with specific characteristics or experiences relevant to the research. This method proves invaluable when researchers seek in-depth insights from particular groups. Students should appreciate the strategic nature of purposive sampling, recognizing its role in qualitative research's nuanced exploration. By understanding the intentional selection criteria, students can leverage purposive sampling to gather rich, context-specific data, contributing depth to their research findings. As with any sampling technique, transparency and a clear justification for the chosen approach are critical to maintaining the research's integrity and validity.

Overcoming Sampling Challenges

Despite the diverse array of sampling techniques, challenges persist. Overcoming these challenges requires a multifaceted approach. Researchers must proactively address issues such as sampling bias, non-response, and ethical considerations to ensure the robustness of their findings. Developing strategies to minimize bias, calculating an appropriate sample size, and employing ethical sampling practices are essential components of overcoming sampling challenges. Moreover, understanding the unique hurdles associated with each sampling technique, such as the potential biases in non-random methods like snowball and convenience sampling, empowers researchers to make informed choices. As students engage with assignments, mastering these strategies will not only enhance the quality of their research but also equip them with valuable skills for addressing sampling challenges in diverse research scenarios. Embracing a proactive and informed approach to sampling challenges is key to advancing the accuracy and credibility of survey-based research.

Sampling Bias

Sampling bias, a persistent challenge in research, emerges when the selected sample deviates from accurately representing the entire population. It manifests in various forms, including selection bias and response bias, which students must discern and address in their assignments. To mitigate these biases, students can implement robust strategies, such as employing diverse sampling methods, randomization, and careful participant selection. By understanding and actively working to minimize sampling bias, students enhance the validity and generalizability of their survey-based research, contributing to more reliable academic outcomes.

Sample Size Determination

The determination of an appropriate sample size is a critical aspect influencing the reliability of survey results. Students should delve into statistical methodologies and tools to calculate the optimal sample size. Factors like confidence level, margin of error, and variability within the population demand careful consideration. By acquiring a nuanced understanding of sample size determination, students empower themselves to design studies with statistical precision, ensuring their findings accurately reflect the broader population and reinforcing the robustness of their research endeavors.

Handling Non-Response

Non-response, a prevalent challenge in survey-based research, can introduce bias and compromise the integrity of findings. Students need to proactively learn and implement strategies to minimize non-response bias. This involves conducting follow-up surveys, employing diverse communication channels, and analyzing non-respondents' characteristics to identify potential biases. Through these proactive measures, students can address non-response challenges, bolstering the reliability of their research outcomes and cultivating a more comprehensive understanding of the surveyed population.

Ethical Considerations in Sampling

In the ethical landscape of research, particularly with human subjects, students must navigate a complex terrain. Ethical considerations in sampling are paramount, requiring students to adhere to established guidelines and principles. Maintaining respect for participants' rights and well-being is crucial. By being cognizant of ethical standards, students ensure the integrity of their research, fostering trust in the academic community and contributing to the responsible advancement of knowledge. This ethical awareness becomes an integral aspect of students' research endeavors, shaping not only the validity of their findings but also their commitment to ethical conduct in the scientific pursuit.

In conclusion, mastering advanced sampling techniques is pivotal for students engaged in survey-based research. A solid understanding of various sampling methods, their applications, and potential pitfalls can elevate the quality of assignments and research projects. By delving into the intricacies of sampling, students not only enhance the credibility of their findings but also contribute to the broader scientific community's pursuit of accurate and meaningful knowledge. As students cultivate expertise in these techniques, they empower themselves to navigate the complexities of research, ensuring that their contributions stand as valuable pillars in the edifice of academic knowledge. In this continuous journey of exploration and learning, students become active participants in the evolution of research methodologies, playing a vital role in shaping the future of scientific inquiry and inquiry-driven progress. Ultimately, the investment in mastering advanced sampling techniques is not only an investment in individual academic success but also a contribution to the collective advancement of knowledge across diverse fields and disciplines.

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Conference Acts to Promote Random Case Assignment

Published on March 12, 2024

The Judicial Conference of the United States has strengthened the policy governing random case assignment, limiting the ability of litigants to effectively choose judges in certain cases by where they file a lawsuit.

The policy addresses all civil actions that seek to bar or mandate state or federal actions, “whether by declaratory judgment and/or any form of injunctive relief.” In such cases, judges would be assigned through a district-wide random selection process.

“Since 1995, the Judicial Conference has strongly supported the random assignment of cases and the notion that all district judges remain generalists,” said Judge Robert J. Conrad, Jr., secretary of the Conference. “The random case-assignment policy deters judge-shopping and the assignment of cases based on the perceived merits or abilities of a particular judge. It promotes the impartiality of proceedings and bolsters public confidence in the federal Judiciary.”

In most of the nation’s 94 federal district courts, local case assignment plans facilitate the random selection of judges. Some plans assign cases to a judge in the division of the court where the case is filed. In divisions where only a single judge sits, these rules have made it possible for a litigant to pre-select that judge by filing in that division. 

In a November 2021 letter, Senator Thom Tillis (R-N.C.), and Patrick Leahy, a Vermont senator who since has retired, raised concerns about a concentration of patent cases filed in single-judge divisions. 

Chief Justice John G. Roberts, Jr., referenced this letter in his 2021 Year-End Report on the Federal Judiciary , calling for a study of judicial assignment practices in patent cases.

“Senators from both sides of the aisle have expressed concern that case assignment procedures … might, in effect, enable the plaintiff to select a particular judge to hear a case,” Roberts said. During the patent-case study, the Court Administration and Case Management Committee (CACM) determined that similar issues might occur in bankruptcy and other types of civil litigation. Public debate grew when several highly controversial lawsuits, seeking nationwide injunctions against federal government policies, were filed in single-judge court divisions.

In submitting the proposed policy to the Judicial Conference, the CACM Committee said that some local case assignment plans risked creating an appearance of “judge shopping.” The committee also noted that the value of trying a civil case in the nearest court division becomes less important when the impact of a ruling might be felt statewide or even nationally.

The amended policy applies to cases involving state or federal laws, rules, regulations, policies, or executive branch orders. District courts may continue to assign cases to a single-judge division when they do not seek to bar or mandate state or federal actions, whether by declaratory judgment and/or any form of injunctive relief.

In addition to the Judiciary policy, the CACM committee will disseminate guidance to all district courts regarding civil case assignment.

The  26-member Judicial Conference  is the policy-making body for the federal court system. By statute, the Chief Justice of the United States serves as its presiding officer and its members are the chief judges of the 13 courts of appeals, a district judge from each of the 12 geographic circuits, and the chief judge of the Court of International Trade.

The Conference convenes twice a year to consider administrative and policy issues affecting the court system.

Related Topics:  Judicial Conference of the United States