Let's walk through the solution step by step. We're conducting a hypothesis test on the mean customer rating of a restaurant. Here's how to approach this:

(a) Null Hypothesis and Alternative Hypothesis

We are testing whether the mean customer rating has changed after the restaurant was remodeled. The hypotheses are:

• Null Hypothesis (H₀): The mean customer rating is still 75, meaning the remodeling had no effect. $H₀: \mu = 75$

• Alternative Hypothesis (H₁): The mean customer rating is different from 75 (this implies a two-tailed test). $H₁: \mu \neq 75$

(b) Performing the Hypothesis Test (Z-test)

Step 1: Type of test

Since we are testing whether the mean is different from 75 (either higher or lower), this is a two-tailed test.

Step 2: Calculate the critical value

At the 0.10 significance level for a two-tailed test, the significance level in each tail is 0.05. The critical values for a Z-test at the 0.05 level are approximately ±1.645. These are the critical Z-values that define the rejection regions.

Step 3: Calculate the test statistic

We use the formula for the Z-test: $Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$ Where:

• $\bar{x} = 72.5$ (sample mean)
• $\mu = 75$ (hypothesized population mean)
• $\sigma = 15.6$ (population standard deviation)
• $n = 37$ (sample size)

Now, plug in the values: $Z = \frac{72.5 - 75}{\frac{15.6}{\sqrt{37}}} = \frac{-2.5}{2.562} \approx -0.976$

The test statistic is -0.976.

(c) Conclusion

The calculated Z-test statistic is -0.976, which is within the range of $[-1.645, 1.645]$, meaning it does not fall in the rejection region.

Thus, the correct conclusion is:

• Since the value of the test statistic doesn't lie in the rejection region, the null hypothesis is not rejected. Therefore, there is not enough evidence to support the claim that the mean customer rating is different from 75.

Do you want more details or have any questions?

Here are five questions to further expand your understanding:

1. What is the significance of using a two-tailed test in this problem?
2. How would the conclusion change if we used a different level of significance?
3. What would happen to the test statistic if the sample size increased?
4. How does the population standard deviation affect the Z-test?
5. Why is it important to check whether the population standard deviation is known?

Tip: In hypothesis testing, choosing the correct tail for the test (one-tailed or two-tailed) is essential based on the alternative hypothesis.