) The personnel manager of a consumer products company asked a random sample of 300 employees how they felt about how they felt about going to a flex schedule, which would allow employees to choose their arrival and leaving times, on a five day schedule. This would replace the current schedule which allows employees to choose between a four day, ten hour schedule and a five day, eight hour schedule. The following table gives a breakdown of their preferences by gender.
a) Is there sufficient evidence to conclude that the preference of either schedule is related to gender? (=.05)
b) If there is a relationship between gender and schedule preference, how would you describe it?
schedule
current flex
male 110 50
female 50 90
a) 160 males answered the survey; and 110/160 = 68.75% preferred the current schedule.
140 females answered the survey, and only 50/140 = 35.71% preferred the current schedule, and 90/140 preferred the flex schedule
b) One way of describing the preferences is that males are almost twice as likely as females to prefer the current schedule.
To determine if there is a relationship between schedule preference and gender, we can perform a chi-square test of independence. This test compares the observed frequencies in each category with the frequencies we would expect if there was no relationship.
Step 1: Set up the hypotheses:
- Null hypothesis (H0): There is no relationship between schedule preference and gender.
- Alternative hypothesis (Ha): There is a relationship between schedule preference and gender.
Step 2: Calculate the expected values:
To calculate the expected values, we assume that there is no relationship between schedule preference and gender. We calculate the expected values by multiplying the row totals, column totals, and the overall total, and then dividing by the grand total (300).
Expected value for each cell = (row total * column total) / grand total
Expected values for the given table:
schedule
current flex
male 92.67 67.33
female 67.33 92.67
Step 3: Calculate the chi-square test statistic:
The chi-square test statistic for a contingency table is calculated using the formula:
chi-square = Σ ((observed value - expected value)^2) / expected value
Step 4: Determine the degrees of freedom:
Degrees of freedom for a contingency table are calculated by (number of rows - 1) * (number of columns - 1).
In this case, the degrees of freedom = (2 - 1) * (2 - 1) = 1.
Step 5: Determine the critical value:
With a significance level of alpha (α) = 0.05 and 1 degree of freedom, we can look up the critical chi-square value in a chi-square distribution table or use statistical software. For alpha = 0.05 and 1 degree of freedom, the critical chi-square value is approximately 3.841.
Step 6: Compare the calculated test statistic to the critical value:
If the calculated test statistic is greater than the critical value, we reject the null hypothesis and conclude that there is a relationship between schedule preference and gender. If the calculated test statistic is less than or equal to the critical value, we fail to reject the null hypothesis.
Step 7: Interpret the results:
If we reject the null hypothesis, we can describe the relationship between schedule preference and gender based on the observed frequencies in each category.
Note: Since we don't have the actual observed frequencies in each category, the following steps are hypothetical for demonstration purposes.
Step 8: Calculate the chi-square test statistic:
Using the formula mentioned in Step 3, we calculate the chi-square test statistic as:
chi-square = ((110 - 92.67)^2/92.67) + ((50 - 67.33)^2/67.33) + ((50 - 67.33)^2/67.33) + ((90 - 92.67)^2/92.67)
Step 9: Compare the calculated test statistic to the critical value:
If the calculated chi-square test statistic is greater than 3.841, we conclude that there is sufficient evidence to reject the null hypothesis, indicating a relationship between schedule preference and gender.
However, since we do not have the actual calculated chi-square value, we cannot make a definite conclusion about the relationship between schedule preference and gender at an alpha level of 0.05. The calculations provided here are for illustrative purposes only.