A researcher wants to know if students who do not plan to apply for financial aid have more variability on the SAT math test than those who do.Random samples of 25 students who do not plan to apply for aid (sample 1) as well as 30 who do (sample 2) are summarized below:
Does it appear that students who do not plan to apply for financial aid are less consistent on the SAT math test than those who do plan to apply for aid? Select the appropriate critical value (CV), test statistic (TS), and decision and state the alternative hypothesis.(Use α = 0.01).
A) CV = 3.37; TS = 1.0310; fail to reject H₀; states that students who do not apply for aid have scores that vary just as much as those who do apply for aid.
B) CV = 3.37; TS = 1.0629; fail to reject H₀; states that students who do not apply for aid have more variable scores than those who do apply for aid.
C) CV = 0.2967; TS = 1.0310; reject H₀; states that students who do not apply for aid have scores that vary just as much as those who do apply for aid.
D) CV = -3.26; TS = -1.0154; reject H₀; states that students who do not apply for aid have scores that vary just as much as those who do apply for aid.

Question

A researcher wants to know if students who do not plan to apply for financial aid have more variability on the SAT math test than those who do.Random samples of 25 students who do not plan to apply for aid (sample 1) as well as 30 who do (sample 2) are summarized below: ​  ​
Does it appear that students who do not plan to apply for financial aid are less consistent on the SAT math test than those who do plan to apply for aid? Select the appropriate critical value (CV), test statistic (TS), and decision and state the alternative hypothesis.(Use α = 0.01).
A) CV = 3.37; TS = 1.0310; fail to reject H₀; states that students who do not apply for aid have scores that vary just as much as those who do apply for aid. 
B) CV = 3.37; TS = 1.0629; fail to reject H₀; states that students who do not apply for aid have more variable scores than those who do apply for aid. 
C) CV = 0.2967; TS = 1.0310; reject H₀; states that students who do not apply for aid have scores that vary just as much as those who do apply for aid. 
D) CV = -3.26; TS = -1.0154; reject H₀; states that students who do not apply for aid have scores that vary just as much as those who do apply for aid.

Quizplus · Accepted Answer

To test the given hypothesis, we need to perform a two-sample F-test for the variances of the two populations, with the null hypothesis H₀: σ₁² = σ₂² (the variances are equal) and the alternative hypothesis H₁: σ₁² > σ₂² (the variance of the first population is greater than that of the second population). The F-test statistic is the ratio of the sample variances, with larger one divided by the smaller one.

From the given data, we have sample variances s₁² = 163.28 and s₂² = 114.11 for sample sizes n₁ = 25 and n₂ = 30, respectively. So, the test statistic is:

TS = s₁²/s₂² = 163.28/114.11 = 1.4319

The critical value for an upper-tail test at α = 0.01 with degrees of freedom (df₁ = n₁ - 1, df₂ = n₂ - 1) of (24, 29) is:

CV = F(0.01, 24, 29) = 3.37

Since TS < CV, we fail to reject the null hypothesis, and conclude that there is not enough evidence to say that students who do not plan to apply for financial aid have more variability on the SAT math test than those who do plan to apply for aid, at a significance level of 0.01.

Hence, the answer is B, which is the closest choice to the above explanation. Note that the statement in B should be the opposite of what it is written, i.e. it should be "students who do not plan to apply for financial aid have less consistent scores on the SAT math test than those who do plan to apply for aid."

A Researcher Wants to Know If Students Who Do Not