Let's work through the problem step by step.

Given information:

• Population mean ($\mu$) = 47.5
• Population standard deviation ($\sigma$) = 12.6
• Sample size ($n$) = 112

We are dealing with the sample mean ($\overline{x}$), and according to the Central Limit Theorem, for a large enough sample size, the distribution of the sample mean will be approximately normal with:

• Mean of sample means ($\mu_{\overline{x}}$) = population mean ($\mu = 47.5$)
• Standard error of the mean ($\text{SE}_{\overline{x}}$) = $\frac{\sigma}{\sqrt{n}} = \frac{12.6}{\sqrt{112}}$

Let's now solve the parts:

Part (a) – Finding the probability that $\overline{x}$ is between 45 and 48:

1. Calculate the standard error (SE): $\text{SE}_{\overline{x}} = \frac{12.6}{\sqrt{112}} \approx 1.19$

2. Find Z-scores for the values 45 and 48:

   • For $\overline{x} = 45$: $Z = \frac{45 - 47.5}{1.19} \approx -2.10$
   • For $\overline{x} = 48$: $Z = \frac{48 - 47.5}{1.19} \approx 0.42$

3. Look up the corresponding probabilities from the standard normal table:

   • $P(Z < -2.10) \approx 0.0179$
   • $P(Z < 0.42) \approx 0.6628$

4. Find the probability that $\overline{x}$ is between 45 and 48: $P(45 < \overline{x} < 48) = P(Z < 0.42) - P(Z < -2.10) = 0.6628 - 0.0179 = 0.6449$ So, the probability is approximately 0.645 (or 64.5%).

Part (b) – Would it be unusual for $\overline{x}$ to be greater than 48?

To determine if it is unusual, we can calculate the probability that $\overline{x}$ is greater than 48:

1. Find the Z-score for $\overline{x} = 48$ (calculated above): $Z = 0.42$

2. Find the probability that $\overline{x}$ is greater than 48: $P(\overline{x} > 48) = 1 - P(Z < 0.42) = 1 - 0.6628 = 0.3372$

So, the probability that $\overline{x} > 48$ is approximately 33.72%. Since this probability is not very small (usually "unusual" events are considered to have a probability below 5%), it would not be unusual for $\overline{x}$ to be greater than 48.

Do you want more details or have any questions?
Here are some related questions to consider:

1. What is the meaning of a Z-score in this context?
2. How would the probability change if the sample size were smaller?
3. Can we apply the same approach if the population were not normally distributed?
4. What is the impact of increasing the population standard deviation on the results?
5. Why is the Central Limit Theorem important for this type of problem?

Tip: Always check the sample size. For small sample sizes, you may need to use the t-distribution instead of the normal distribution.