The executives of CareFree Insurance,Inc.feel that "a majority of our employees perceive a participatory management style at CareFree." A random sample of 200 CareFree employees is selected to test this hypothesis at the 0.05 level of significance. Eighty employees rate the management as participatory. The appropriate decision is __________.
A) fail to reject the null hypothesis
B) reject the null hypothesis
C) reduce the sample size
D) increase the sample size
E) do nothing

Question

Quizplus · Accepted Answer

To test the hypothesis, we need to conduct a hypothesis test. Let's assume the null hypothesis is "less than or equal to 50% of CareFree employees perceive a participatory management style", and the alternative hypothesis is "more than 50% of CareFree employees perceive a participatory management style." 
Using a one-sample proportion test with a significance level of 0.05, we calculate the test statistic as follows:

z = (p̂ - p0) / √(p0(1-p0) / n)

where p̂ is the sample proportion (80/200 = 0.4), p0 is the hypothesized population proportion (0.5), and n is the sample size (200).

z = (0.4 - 0.5) / √(0.5(1-0.5) / 200) = -2.82

The corresponding p-value for this test statistic is less than 0.05, which means we reject the null hypothesis. Therefore, we can conclude that more than 50% of CareFree employees perceive a participatory management style.

Choices C, D, and E are not appropriate because changing the sample size will not affect our conclusion in this case.

The Executives of CareFree Insurance,Inc