In Exercises 5–20, conduct the hypothesis test and provide the...

Question

In Exercises 5–20, conduct the hypothesis test and provide the test statistic and the P-value and/or critical value, and state the conclusion.

Bias in Clinical Trials? Researchers investigated the issue of race and equality of access to clinical trials. The following table shows the population distribution and the numbers of participants in clinical trials involving lung cancer (based on data from “Participation in Cancer Clinical Trials,” by Murthy, Krumholz, and Gross, Journal of the American Medical Association, Vol. 291, No. 22). Use a 0.01 significance level to test the claim that the distribution of clinical trial participants fits well with the population distribution. Is there a race/ethnic group that appears to be very underrepresented?

Accepted Answer

All right, hello, everyone. So this question says, a university is investigating whether the distribution of students participating in a new leadership program matches the distribution of the student body by major. The table below shows the percentage of students in each major and the number of students from each major who participated in the program. At the 0.01 significance level, test the claim that the program participants are representative of the overall student body. Which major appears to be the most underrepresented? All right, so first, the hypotheses relevant to this context are based around whether or not the distribution of the program matches that of the student body. So the null hypothesis, that's H not would state that the distribution of program participants matches that of the student body. The alternative hypothesis, on the other hand, would state that the distributions do not match. So it's just the opposite of the null. But before we can do anything, we have to understand the total number of participants in this leadership program. And you do so by adding together. All of The people in each major that participated in the program. So the total. Would be the sum of 85, 48, 1018, and 2, which gives you 163 people. So from here, we move on to calculating the expected amount of people from each major, based on the student body percentages which we do have. So for business, for example. You would multiply the total number of people participating in this program. By the percentage of business majors as a decimal. So 163 multiplied by 0.40. Would give you 65.20. That's the total number of people you would expect to see studying business in this program. And so now you would repeat the same calculation for your remaining majors, that's engineering, arts, science, and education. When you do so, the expected counts are as follows. For engineering, that's 40.75. For arts and science, that would be 24.45. And for education, that would be 8.15. So now it's this information that's going to be relevant for the chi square test statistic. So Here we would use chi square is equal to the sum of. The differences between the observed and expected frequencies for each major squared, divided by the expected frequency. So here's how it's going to go, right? The data that's already provided to us represents the observed frequencies for each major. What we just calculated is the expected frequency, given the percentage of people studying a given major at the university. So for each data point, right, for each major, you're going to calculate the following. First, you need to know the difference between the observed and the expected frequency, again for each major. And then you're going to take that value that is the difference between observed and expected and square it. And then for each major, you're going to take the difference between observed and expected squared. And divide that by the expected frequency. So let me demonstrate this point with the information given for business. So for the business majors, the observed frequency is 85, because that's the number of people in the program. And the expected frequency was 65.20. The difference between those values is 19.8. 19.8 squared is 392.04. And 392.04 divided by 65.20, which is the expected frequency for business majors, is 6.01. So then you're going to repeat this calculation for the remaining majors. And when you do so, The sum of. The sum of each value. For the difference between observed and expected squared divided by expected. Equals 22.18. And it's 22.18, that is your test statistic. So now The next step is to find the critical value. So for the critical value, recall that the degrees of freedom is equal to the sample size subtracted by one. In this case, the sample size is 5 majors, which subtracted by 1. Gives you 4 degrees of freedom. So, at alpha equals 0.01, the critical value for 4 degrees of freedom. Is approximately 13.277. Which now leads you to your decision. Right, in this case, our test statistic of 22.18 is greater than the critical value, which means that we reject the null hypothesis. So that addresses the first part of the question. Now, which one is the most underrepresented major? Well, to understand this, you would first compare the values for observed versus expected, and then you would pick the major that has the greatest difference between them. Focusing on the arts, for example. There are 10 people in the program studying arts, and the expected frequency is 24.45. This gives you a difference of 14.45, which happens to be the greatest out of all the majors represented in this program. So because the arts major has the largest shortfall, the largest difference between observed and expected, it is the most underrepresented. So now this leads you to the final answer, which reads as follows. At the 0.01 significance level, there is sufficient evidence to support the claim. That the program participants are not representative of the student body. And the arts major appears to be the most underrepresented. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

Key Concepts

Hypothesis Testing

P-value

Chi-Square Test for Goodness of Fit

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