In a recent marathon in Boston, 24,521 men finished and 283 dropped out. Also 12

Question

In a recent marathon in Boston, 24,521 men finished and 283 dropped out. Also 12,853 women
finished and 152 dropped out. Use a 0.05 significance level to test the claim that the rate of
those who finished is the same for men and women.
a. State the null and alternative hypothesis

b. Manually calculate the test statistic and find the critical value
c. Test the claim by constructing an appropriate confidence interval.
d. Based on the results, do men and women finish the Boston marathon at the same rate
(proportion)?

Explanation / Answer

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: P1 = P2

Alternative hypothesis: P1 P2

Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the proportion from population 1 is too big or if it is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a two-proportion z-test.

Analyze sample data. Using sample data, we calculate the pooled sample proportion (p) and the standard error (SE). Using those measures, we compute the z-score test statistic (z).

p = (p1 * n1 + p2 * n2) / (n1 + n2)

p = 0.01164

SE = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }

SE = 0.001168

z = (p1 - p2) / SE

z = - 0.24

zcritical = + 1.96

where p1 is the sample proportion in sample 1, where p2 is the sample proportion in sample 2, n1 is the size of sample 1, and n2 is the size of sample 2.

Since we have a two-tailed test, the P-value is the probability that the z-score is less than - 0.24 or greater than 0.24.

Thus, the P-value = 0.8104.

Interpret results. Since the P-value (0.8104) is greater than the significance level (0.05), we have to accept the null hypothesis.

From the above test we have sufficient evidence in the favor of the claim that men and women finish the Boston marathon at the same rate.

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