The post office, which has separate lines at different windows, determines that the standard deviation for customer waiting times on Friday afternoons is 7.2 minutes. The post office decides to experiment with a single, main waiting line. For a random sample of 25 customers, it is found that the waiting times have a standard deviation of 3.5 minutes on a Friday afternoon. At a 5% significance level, the post office wants to test the assertion that a single line results in less variation in waiting times for customers. Give your conclusion.
Question
The post office, which has separate lines at different windows, determines that the standard deviation for customer waiting times on Friday afternoons is 7.2 minutes. The post office decides to experiment with a single, main waiting line. For a random sample of 25 customers, it is found that the waiting times have a standard deviation of 3.5 minutes on a Friday afternoon. At a 5% significance level, the post office wants to test the assertion that a single line results in less variation in waiting times for customers. Give your conclusion.
Solution
To test the assertion that a single line results in less variation in waiting times for customers, we can perform a hypothesis test for the difference in standard deviations.
Step 1: State the hypotheses. The null hypothesis is that the standard deviation of the waiting times with a single line is equal to the standard deviation of the waiting times with separate lines. The alternative hypothesis is that the standard deviation of the waiting times with a single line is less than the standard deviation of the waiting times with separate lines.
Step 2: Formulate an analysis plan. For this analysis, the significance level is defined as 0.05. We will use a chi-square test to compare the standard deviations.
Step 3: Analyze sample data. Using the sample data, we calculate the chi-square test statistic (x^2) using the formula: x^2 = (n-1)s^2 / σ^2. Where s is the sample standard deviation (3.5 minutes), n is the sample size (25 customers), and σ is the population standard deviation (7.2 minutes).
Step 4: Interpret the results. If the p-value is less than the significance level (0.05), we reject the null hypothesis in favor of the alternative hypothesis. This would suggest that a single line results in less variation in waiting times for customers.
Without the actual calculations and p-value, we can't give a definitive conclusion. However, this is the process you would follow to test the assertion.
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