Problem

A human resources representative claims that the proportion of employees earning more than $\$ 50,000$ is less than $40 \%$. To test this claim, a random sample of 700 employees is taken and 305 employees are determined to earn more than $\$ 50,000$. The following is the setup for this hypothesis test: \[ \begin{array}{l} H_{0}: p=0.40 \\ H_{a}: p<0.40 \end{array} \] In this example, the $p$-value was determined to be 0.973 . Come to a conclusion and interpret the results for this hypothesis test for a proportion (use a significance level of $5 \%$ )

Solution

Step 1 :The human resources representative claims that the proportion of employees earning more than $50,000 is less than 40%. To test this claim, a random sample of 700 employees is taken and 305 employees are determined to earn more than $50,000. The null hypothesis (H0) is that the proportion of employees earning more than $50,000 is equal to 40%, and the alternative hypothesis (Ha) is that the proportion is less than 40%.

Step 2 :The p-value for this hypothesis test is calculated to be 0.973. The p-value is the probability of obtaining the observed data or more extreme data, given that the null hypothesis is true.

Step 3 :We compare the p-value to the significance level of 5%. If the p-value is less than the significance level, we reject the null hypothesis. If the p-value is greater than the significance level, we do not reject the null hypothesis.

Step 4 :In this case, the p-value (0.973) is greater than the significance level (0.05), so we do not reject the null hypothesis.

Step 5 :\(\boxed{\text{Final Answer: We cannot reject the null hypothesis. Therefore, we cannot conclude that the proportion of employees earning more than $50,000 is less than 40%}}\)

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Source: https://solvelyapp.com/problems/16604/

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