To test whether the standard deviation of waiting times in the single waiting line setup is less than 2.2 minutes, we will perform a hypothesis test for a population standard deviation using a chi-square distribution. The steps are outlined as follows:

(a) State the Hypotheses:

1. Null Hypothesis (H₀): The standard deviation of waiting times, $\sigma$, is equal to or greater than 2.2 minutes. $H₀: \sigma \geq 2.2$

2. Alternative Hypothesis (H₁): The standard deviation of waiting times, $\sigma$, is less than 2.2 minutes. $H₁: \sigma < 2.2$

(b) Determine the Significance Level and Identify the Critical Value:

• Given the significance level, $\alpha = 0.05$.
• We will use a chi-square distribution for this test since it is based on the sample variance.

Let:

• $n$ = sample size (number of customers in the data table).
• $s$ = sample standard deviation of waiting times (calculated from the data).

The test statistic for testing the population variance $\sigma^2$ is: $\chi^2 = \frac{(n - 1)s^2}{\sigma^2}$ where $\sigma^2 = 2.2^2 = 4.84$.

To find the critical value, we use the chi-square distribution table for $n - 1$ degrees of freedom at the 0.05 significance level for a left-tailed test.

(c) Calculate the Test Statistic:

Using the sample data provided, calculate the sample standard deviation $s$ and then apply it in the formula for the test statistic $\chi^2$.

(d) Make a Decision:

1. If the calculated $\chi^2$ is less than the critical value from the chi-square table, we reject the null hypothesis.
2. Otherwise, we fail to reject the null hypothesis.

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If you can provide the data in the table or the sample size and standard deviation calculated from it, I can assist with the actual computations.

Would you like details on calculating the sample standard deviation or interpreting chi-square critical values?

Follow-Up Questions:

1. What is the procedure to calculate the sample standard deviation from raw data?
2. How does the chi-square distribution differ for left-tailed and right-tailed tests?
3. Why do we use the chi-square test for population variance instead of other tests?
4. How is the decision affected if we change the significance level to a higher or lower value?
5. How does the interpretation change if the sample is not normally distributed?

Tip: When using the chi-square test, always check if the sample data meets the normality assumption, as this affects test validity.