From the screenshot, the task involves hypothesis testing. Here's a breakdown of the steps you need to follow:

(a) State the null hypothesis $H_0$ and the alternative hypothesis $H_a$:

1. Null Hypothesis $H_0$: The mean customer rating is equal to 65 (status quo). $H_0: \mu = 65$
2. Alternative Hypothesis $H_a$: The mean customer rating is greater than 65 (claim by management). $H_a: \mu > 65$

(b) Perform the hypothesis test:

The test statistic is calculated using: $z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$ Where:

• $\bar{x} = 76.2$ (sample mean),
• $\mu = 65$ (population mean under $H_0$),
• $\sigma = 22.8$ (population standard deviation),
• $n = 32$ (sample size).

Step 1: Calculate $z$:

Substitute the values into the formula: $z = \frac{76.2 - 65}{\frac{22.8}{\sqrt{32}}}$

1. Calculate the denominator: $\frac{22.8}{\sqrt{32}} = \frac{22.8}{5.657} \approx 4.031$
2. Calculate $z$: $z = \frac{76.2 - 65}{4.031} = \frac{11.2}{4.031} \approx 2.78$

Step 2: Determine the critical value:

• For a one-tailed test at the 0.10 significance level ($\alpha = 0.10$), the critical value from the standard normal distribution table is approximately $z = 1.28$.

Step 3: Compare $z$ with the critical value:

• Since $z = 2.78 > 1.28$, we reject the null hypothesis $H_0$.

(c) Conclusion:

At the 0.10 level of significance, there is enough evidence to support the claim that the mean customer rating is greater than 65.

Do you want me to explain any specific part of the solution or assist with further calculations?
Here are 5 related questions for deeper understanding:

1. What happens if the significance level changes to 0.05?
2. How is the critical value determined for a two-tailed test?
3. What assumptions are necessary for using the $z$-test in hypothesis testing?
4. How does increasing the sample size affect the test statistic?
5. Can the population standard deviation always be assumed to be known?

Tip: Always verify whether the population standard deviation ($\sigma$) is given, as this determines whether to use the (z