Problem Setup:

We are tasked with investigating whether the population mean $\mu$ of satisfaction ratings exceeds 42, based on a sample of 65 customer satisfaction ratings. Specifically, we want to find the probability of observing a sample mean $\bar{X} \geq 42.95$ under the null hypothesis $H_0: \mu = 42$.

Given:

• $\mu_0 = 42$ (null hypothesis mean)
• $\bar{X} = 42.95$ (sample mean)
• $\sigma = 2.64$ (population standard deviation)
• $n = 65$ (sample size)

Step 1: Standard Error of the Mean

The standard error (SE) of the sample mean $\bar{X}$ is calculated using the formula:

$SE = \frac{\sigma}{\sqrt{n}} = \frac{2.64}{\sqrt{65}}$

Let's calculate that.

Step 2: Z-Score Calculation

To assess the probability, we need to compute the z-score for the sample mean $\bar{X} = 42.95$. The z-score formula is:

$z = \frac{\bar{X} - \mu_0}{SE}$

Where $\mu_0 = 42$. We can now compute this.

Step 3: Find the Probability

Once we have the z-score, we will use the standard normal distribution to find the probability of observing a z-score greater than or equal to this value.

I'll calculate each step.### Calculations:

1. Standard Error (SE):
   $SE = 0.3275$

2. Z-Score:
   $z = 2.90$

3. Probability (P-value):
   The probability of observing a sample mean $\bar{X} \geq 42.95$ when $\mu_0 = 42$ is approximately: $P(\bar{X} \geq 42.95) = 0.0019$

Since the P-value is very small (less than 0.05), this indicates that the sample provides strong evidence against the null hypothesis, suggesting that the population mean $\mu$ is likely greater than 42.

Would you like further details or have any questions?

Here are 5 related questions:

1. What is the interpretation of a low P-value in hypothesis testing?
2. How does increasing the sample size affect the standard error?
3. Why is it important to know the population standard deviation in hypothesis testing?
4. What would happen if we assumed a different value for the population mean?
5. How would you interpret this result if the P-value was higher than 0.05?

Tip: Always ensure that assumptions about population parameters (such as the known standard deviation) are justified when conducting hypothesis tests.