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)do not 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

We need to conduct a hypothesis test to determine whether the proportion of employees who perceive a participatory management style at CareFree is greater than 0.5 (since "a majority" indicates more than 50%).

Let p be the true population proportion of employees who perceive a participatory management style at CareFree.

Our null hypothesis is:
H0: p = 0.5

Our alternative hypothesis is:
Ha: p > 0.5

We will use a one-tailed z-test to test this hypothesis, since the alternative hypothesis is one-sided. The test statistic is:
z = (p̂ - 0.5) / sqrt(0.5 * 0.5 / n),
where p̂ is the sample proportion, n is the sample size, and sqrt means "square root".

In this case, p̂ = 80/200 = 0.4, n = 200:

z = (0.4 - 0.5) / sqrt(0.5 * 0.5 / 200) = -2.83

The critical value for a one-tailed test at the 0.05 level of significance is:
zα = 1.645

Since -2.83 < -1.645, we reject the null hypothesis and conclude that there is sufficient evidence to support the claim that less than a majority of CareFree employees perceive a participatory management style.

Therefore, the appropriate decision is to reject the null hypothesis (choice B).

The Executives of CareFree Insurance, Inc