To address the questions in this problem, let's break down each statement and determine the type of statistical test or calculation we need to perform:

1. Statement: "How likely it is in a sample of 250 we will find that the mean number of hours per week corporate employees work is as high as 47."

   This question is asking for the probability that a sample mean of 47 hours could occur, assuming we have information about the population mean and standard deviation.

2. Statement: "How likely it is that the true mean number of hours per week corporate employees work is 40."

   This question is asking us to evaluate the likelihood that the population mean is exactly 40 hours. This is generally framed as a hypothesis test, with the null hypothesis that the mean number of hours worked is 40.

3. Statement: "How likely it is that the true mean number of hours per week corporate employees work is more than 40."

   Here, we are evaluating if the evidence (sample mean of 47) supports the claim that the true mean is more than 40 hours. This is also a hypothesis test but with a one-sided alternative hypothesis, where the null hypothesis is that the mean is 40 hours, and the alternative hypothesis is that the mean is greater than 40.

4. Statement: "How likely it is that in a sample of 250 we will find that the mean number of hours per week corporate employees work is as high as 47 if the true mean is 40."

   This question is looking for the probability of obtaining a sample mean of 47, assuming the true population mean is 40. We would use the sampling distribution of the mean to calculate this probability, often using the Z-score formula.

Approach to Solution

Given:

• Sample mean ($\bar{x}$) = 47 hours
• Population standard deviation ($\sigma$) = 3.2 hours
• Sample size (n) = 250
• Hypothetical population mean ($\mu$) = 40 (for testing purposes)

Using this information, we can calculate the Z-score for a sample mean of 47 under the assumption that the true population mean is 40.

Z-score Formula

$Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$ Where:

• $\bar{x}$ is the sample mean,
• $\mu$ is the population mean (40 in this case),
• $\sigma$ is the standard deviation of the population,
• $n$ is the sample size.

Let's calculate the Z-score for a sample mean of 47 hours.

Would you like to proceed with these calculations, or do you need further explanation on any specific part?

Here are five questions that could further your understanding:

1. What does a Z-score represent in the context of a sampling distribution?
2. How do we interpret a high or low p-value in hypothesis testing?
3. What is the difference between a one-sided and two-sided hypothesis test?
4. How does sample size affect the precision of a sample mean as an estimate of the population mean?
5. Why do we use the standard deviation of the sample mean (standard error) in these calculations?

Tip: Always check whether your test requires a one-tailed or two-tailed approach based on the claim being tested.