An employer polled its employers to see if they agree with the proposed new store hours and whether or not their present shift made a difference in their answers. The customers answered 1 for agree, 2 for don't know, and 3 for disagree. Nine first shift employees answered "agree", 15 second shift employees answered "agree", and 20 third shift employees answered agree. With a 95% confidence level determine whether or not the employees' present shift played a role in their responses to the poll.
To determine whether the employees' present shift played a role in their responses to the poll, we can perform a hypothesis test using the chi-square test for independence. Here's how to calculate it step by step:
Step 1: Formulate the null and alternative hypotheses:
- Null Hypothesis (H0): There is no association between the employees' present shift and their responses to the poll.
- Alternative Hypothesis (H1): There is an association between the employees' present shift and their responses to the poll.
Step 2: Set the significance level (alpha):
- The given problem does not specify the significance level (alpha), so let's assume it to be 0.05 (5%).
Step 3: Calculate the expected frequencies:
- To perform the chi-square test, we need to calculate the expected frequencies for each cell in the contingency table. The expected frequency for each cell is calculated using the formula:
Ei = (Row Total * Column Total) / Grand Total
Here's the calculation for each cell:
| Agree | Don't Know | Disagree | Row Total
First | ? | ? | ? | 9
Second | ? | ? | ? | 15
Third | ? | ? | ? | 20
Column Total: | ? | ? | ? | ?
Step 4: Calculate the chi-square statistic:
- The chi-square statistic is calculated using the formula:
X^2 = ∑ ((Oi - Ei)^2 / Ei)
Here, Oi represents the observed frequencies, and Ei represents the expected frequencies.
Step 5: Determine the degrees of freedom (df):
- The degrees of freedom can be calculated using the formula:
df = (r - 1) * (c - 1)
Where r is the number of rows in the contingency table, and c is the number of columns.
Step 6: Compare the chi-square statistic with the critical value:
- The critical value can be obtained from the chi-square distribution table at the appropriate degrees of freedom and significance level (alpha). We compare the calculated chi-square statistic with the critical value to determine whether to reject or fail to reject the null hypothesis.
If the calculated chi-square statistic is greater than the critical value, we reject the null hypothesis and conclude that there is a relationship between the employees' present shift and their responses.
If the calculated chi-square statistic is less than the critical value, we fail to reject the null hypothesis and conclude that there is no relationship between the employees' present shift and their responses.
You can follow these steps to perform the calculations and determine the statistical significance of the relationship between the employees' present shift and their responses to the poll.