The senior executives of CareFree Insurance, Inc.feel that "a minority of our employees perceive an authoritarian management style at CareFree." A random sample of 200 CareFree employees is selected to test this hypothesis.Eighty employees rate the management as authoritarian.Using $\alpha$ = 0.05, the appropriate decision is ___.

Question

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This is a one-tailed test since we are testing if the proportion of employees who perceive an authoritarian management style is less than 0.5. We are given:
- Sample size n = 200
- Number of employees who rate management as authoritarian X = 80
- Level of significance alpha = 0.05
- Null hypothesis H0: p >= 0.5 (majority of employees do not perceive an authoritarian management style)
- Alternative hypothesis Ha: p < 0.5 (minority of employees perceive an authoritarian management style)

We can use a z-test to test this hypothesis:
z = (X - np) / sqrt(np(1-p))
where p is the proportion of employees who perceive an authoritarian management style under the null hypothesis, which is 0.5.

The critical value for a one-tailed test at alpha = 0.05 is -1.645. 
If z is less than or equal to -1.645, we reject the null hypothesis. Otherwise, we do not reject the null hypothesis.

Calculating the test statistic:
p-hat = X / n = 80 / 200 = 0.4
z = (80 - 100*0.5) / sqrt(100*0.5*0.5) = -2.828

Since -2.828 < -1.645, we reject the null hypothesis and conclude that there is enough evidence to suggest that a minority of employees perceive an authoritarian management style at CareFree.

The Senior Executives of CareFree Insurance, Inc α\alphaα = 005, the Appropriate Decision Is ___

The Senior Executives of CareFree Insurance, Inc $\alpha$ = 005, the Appropriate Decision Is ___