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The press release titled "Keeping Score When It counts: Graduation Rates and Academic Progress Rates" (The Institute for Diversity and Ethics in Sport, March 16,2009 ) gave the 2009 graduation rates for African-American basketball players and for white basketball players at every NCAA Division I university with a basketball program. Explain why it is not necessary to use a paired \(t\) test to determine if the mean graduation rate for African-American basketball players differs from the mean graduation rate for white basketball players for Division I schools.

Short Answer

Expert verified
A paired t-test is not necessary here because graduation rates for African-American basketball players and for white basketball players at NCAA Division I schools are independent groups, and there is no pairing or relationship between an individual in one group with an individual in the other group. For independent groups, an independent t-test would be more appropriate.

Step by step solution

01

Understand the Role of Paired t-test

A paired t-test is used to compare the means of the same group or item under two separate scenarios. In other words, it's used when the observations are dependent. Examples include a scenario where you measure attributes (like weight, blood pressure etc.) of the same individuals before and after a particular treatment.
02

Analyze the Scenario

In this case, you're examining two separate groups: African-American basketball players and white basketball players at NCAA Division I schools. These two groups are independent. The graduation rate of an African-American player does not affect or is not paired with a white player's graduation rate.
03

Conclusion

Because the groups are independent, there is no pairing or connection between the graduation rates of the two different groups. This makes a paired t-test unnecessary in this case. Instead an independent t-test would be more appropriate as it is used to compare the means of two independent groups.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independent t-test
When studying statistical tests, it's important to distinguish between different methods and know when to apply each one. An independent t-test, also known as an unpaired t-test, is a fundamental statistical tool used to determine if there are any statistically significant differences between the means of two unrelated groups.

For example, if we wanted to compare the academic performance of two distinct groups of students from different schools, an independent t-test would help us determine if the observed difference in grades between the two groups is likely to have occurred by chance, or if there is a meaningful difference. Unlike the paired t-test, which looks at related samples, the independent t-test operates under the assumption that the two groups are separate entities, and there is no natural pairing among the elements of the two datasets.

To properly conduct an independent t-test, we need to ensure that data points from one group do not influence or connect to data points from another. The groups should be randomly sampled and their variances should be approximately equal for the test to be valid. This way, any conclusion derived from the t-test is based on the individual characteristics of each group, without any confounding overlap.
Mean Comparison
At the heart of many statistical analyses lies mean comparison. It's the process of evaluating whether the average values (means) from two different sets of data are significantly different from one another. This is incredibly useful across many fields, such as psychology, medicine, economics, and obviously, education.

To compare means effectively, we typically use a variety of t-tests, which help us to understand if any observed differences in sample means are statistically significant, or just due to random variation. For example, to see if a new teaching method is more effective than the current one, comparing the average test scores from each method using a t-test would provide evidence of the method's effectiveness.

Mean comparison becomes even more interesting when we have to account for variance within the groups. In sports, comparing the graduation rates of two cohorts, like NCAA Division I African-American basketball players against their white counterparts, can be insightful; however, we also have to consider within-group factors such as socioeconomic backgrounds or the specific support systems in place at different universities.
Statistical Hypothesis Testing
Statistical hypothesis testing is a core concept in research that allows us to make inferences about populations based on sample data. Typically, we start with a null hypothesis that assumes no effect or no difference, and an alternative hypothesis that contradicts the null. In the context of our example with NCAA graduation rates, the null hypothesis could state that there is no difference in graduation rates between African-American and white basketball players, while the alternative hypothesis would suggest there is a difference.

Through hypothesis testing, such as an independent t-test, we can determine the likelihood that our observed data could have occurred under the null hypothesis. If this likelihood (p-value) is sufficiently low (commonly below 0.05), we reject the null hypothesis in favor of the alternative, suggesting that our findings are statistically significant. This process enables us to draw conclusions from data that go beyond the specific numbers we have calculated, proposing broader patterns or effects that could apply to the general population.
NCAA Division I Graduation Rates
Graduation rates within NCAA Division I institutions are a key performance indicator of both academic success and the effectiveness of athletic programs in supporting student-athletes. These rates reflect the percentage of student-athletes who graduate within a certain period of their initial college enrolment.

Observing disparities in graduation rates between different demographics can prompt necessary conversations and actions regarding equality and support within educational and athletic frameworks. For instance, if empirical data reveal that African-American basketball players have lower graduation rates at Division I schools compared to their white peers, policymakers and educators might need to investigate the underlying causes and potential solutions. This could include access to academic resources, mentorship programs, or socio-economic factors that might influence educational outcomes.

It’s essential to approach the analysis of such sensitive data with robust statistical tools that acknowledge the independence of the populations in question. Only through careful and rigorous analysis can we ensure that any conclusions drawn are reliable and can form the basis for making positive changes in the collegiate athletic community.

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Most popular questions from this chapter

Two proposed computer mouse designs were compared by recording wrist extension in degrees for 24 people who each used both mouse types ("Comparative Study of Two Computer Mouse Designs," Cornell Human Factors Laboratory Technical Report RP7992). The difference in wrist extension was computed by subtracting extension for mouse type \(\mathrm{B}\) from the wrist extension for mouse type A for each student. The mean difference was reported to be 8.82 degrees. Assume that it is reasonable to regard this sample of 24 people as representative of the population of computer users. a. Suppose that the standard deviation of the differences was 10 degrees. Is there convincing evidence that the mean wrist extension for mouse type \(A\) is greater than for mouse type B? Use a .05 significance level. b. Suppose that the standard deviation of the differences was 26 degrees. Is there convincing evidence that the mean wrist extension for mouse type \(A\) is greater than for mouse type B? Use a .05 significance level. c. Briefly explain why a different conclusion was reached in the hypothesis tests of Parts (a) and (b).

Public Agenda conducted a survey of 1379 parents and 1342 students in grades \(6-12\) regarding the importance of science and mathematics in the school curriculum (Associated Press, February \(15,2 \mathrm{OO} 6\) ). It was reported that \(50 \%\) of students thought that understanding science and having strong math skills are essential for them to succeed in life after school, whereas \(62 \%\) of the parents thought it was crucial for today's students to learn science and higher-level math. The two samples - parents and students -were selected independently of one another. Is there sufficient evidence to conclude that the proportion of parents who regard science and mathematics as crucial is different than the corresponding proportion for students in grades \(6-12 ?\) Test the relevant hypotheses using a significance level of .05 .

Two different underground pipe coatings for preventing corrosion are to be compared. The effect of a coating (as measured by maximum depth of corrosion penetration on a piece of pipe) may vary with depth, orientation, soil type, pipe composition, etc. Describe how an experiment that filters out the effects of these extraneous factors could be carried out.

In the experiment described in the paper "Exposure to Diesel Exhaust Induces Changes in EEG in Human Volunteers" (Particle and Fibre Toxicology [2007])\(, 10\) healthy men were exposed to diesel exhaust for 1 hour. A measure of brain activity (called median power frequency, or MPF) was recorded at two different locations in the brain both before and after the diesel exhaust exposure. The resulting data are given in the accompanying table. For purposes of this example, assume that it is reasonable to regard the sample of 10 men as representative of healthy adult males. $$ \begin{array}{ccrcr} \hline & \text { Location 1 } & \text { Location 1 } & \text { Location 2 } & \text { Location 2 } \\ \text { Subject } & \text { Before } & \text { After } & \text { Before } & \text { After } \\ \hline 1 & 6.4 & 8.0 & 6.9 & 9.4 \\ 2 & 8.7 & 12.6 & 9.5 & 11.2 \\ 3 & 7.4 & 8.4 & 6.7 & 10.2 \\ 4 & 8.7 & 9.0 & 9.0 & 9.6 \\ 5 & 9.8 & 8.4 & 9.7 & 9.2 \\ 6 & 8.9 & 11.0 & 9.0 & 11.9 \\ 7 & 9.3 & 14.4 & 7.9 & 9.1 \\ 8 & 7.4 & 11.3 & 8.3 & 9.3 \\ 9 & 6.6 & 7.1 & 7.2 & 8.0 \\ 10 & 8.9 & 11.2 & 7.4 & 9.1 \\ \hline \end{array} $$ a. Do the data provide convincing evidence that the mean MPF at brain location 1 is higher after diesel exposure? Test the relevant hypotheses using a significance level of \(.05 .\) b. Construct and interpret a \(90 \%\) confidence interval estimate for the difference in mean MPF at brain location 2 before and after exposure to diesel exhaust.

After the 2010 earthquake in Haiti, many charitable organizations conducted fundraising campaigns to raise money for emergency relief. Some of these campaigns allowed people to donate by sending a text message using a cell phone to have the donated amount added to their cell-phone bill. The report "Early Signals on Mobile Philanthropy: Is Haiti the Tipping Point?" (Edge Research, 2010 ) describes the results of a national survey of 1526 people that investigated the ways in which people made donations to the Haiti relief effort. The report states that \(17 \%\) of Gen \(Y\) respondents (those born between 1980 and 1988 ) and \(14 \%\) of Gen \(X\) respondents (those born between 1968 and 1979 ) said that they had made a donation to the Haiti relief effort via text message. The percentage making a donation via text message was much lower for older respondents. The report did not say how many respondents were in the Gen \(\mathrm{Y}\) and Gen \(\mathrm{X}\) samples, but for purposes of this exercise, suppose that both sample sizes were 400 and that it is reasonable to regard the samples as representative of the Gen \(\mathrm{Y}\) and Gen \(\mathrm{X}\) populations. a. Is there convincing evidence that the proportion of those in Gen Y who donated to Haiti relief via text message is greater than the proportion for Gen X? Use \(\alpha=.01\). b. Estimate the difference between the proportion of Gen \(\mathrm{Y}\) and the proportion of Gen \(\mathrm{X}\) that made a donation via text message using a \(99 \%\) confidence interval. Provide an interpretation of both the interval and the associated confidence level.

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