The diameter of a brand of tennis balls is approximately normally​ distributed, with a mean of 2.75 inches and a standard deviation of 0.05 inch. A random sample of 12 tennis balls is selected. Complete parts​ (a) through​ (d) below.
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Part 1
a. What is the sampling distribution of the​ mean?
 
A.
Because the population diameter of tennis balls is approximately normally​ distributed, the sampling distribution of samples of size 12 will not be approximately normal.
B.
Because the population diameter of tennis balls is approximately normally​ distributed, the sampling distribution of samples of size 12 cannot be found.
C.
Because the population diameter of tennis balls is approximately normally​ distributed, the sampling distribution of samples of size 12 will be the uniform distribution.
D.
Because the population diameter of tennis balls is approximately normally​ distributed, the sampling distribution of samples of size 12 will also be approximately normal.
Part 2
b. What is the probability that the sample mean is less than 2.74 ​inches? 
​P(Upper X overbarless than2.74​)equals
  
enter your response here
​(Round to four decimal places as​ needed.)
Part 3
c. What is the probability that the sample mean is between 2.73 and 2.77 ​inches?
​P(2.73less thanUpper X overbarless than2.77​)equals
  
enter your response here
​(Round to four decimal places as​ needed.)
Part 4
d. The probability is 65​% that the sample mean will be between what two values symmetrically distributed around the population​ mean?
The lower bound is 
  
enter your response here inches. The upper bound is 
  
enter your response here inches.
​(Round to two decimal places as​ needed.)

Let's go through each part of the problem step by step.

### Part 1: Sampling Distribution of the Mean
The sampling distribution of the sample mean can be understood based on the Central Limit Theorem (CLT). If the population from which samples are drawn is normally distributed, the distribution of the sample mean will also be normally distributed, regardless of the sample size.

Since the problem states that the population diameter of the tennis balls is approximately normally distributed, the sampling distribution of the sample mean for a sample size of 12 will also be approximately normal.

Thus, the correct answer is:
**D. Because the population diameter of tennis balls is approximately normally distributed, the sampling distribution of samples of size 12 will also be approximately normal.**

---

### Part 2: Probability that the Sample Mean is Less Than 2.74 Inches

Here, we need to calculate \( P(\bar{X} < 2.74) \), where \(\bar{X}\) is the sample mean.

- Population mean (\(\mu\)) = 2.75 inches
- Population standard deviation (\(\sigma\)) = 0.05 inches
- Sample size (\(n\)) = 12

The standard error of the mean (SE) is calculated using the formula:
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{0.05}{\sqrt{12}}
\]

We can then calculate the z-score for \( \bar{X} = 2.74 \) using:
\[
z = \frac{\bar{X} - \mu}{SE} = \frac{2.74 - 2.75}{SE}
\]

Once we have the z-score, we can find the probability from the standard normal distribution.

Let's calculate that now.The probability that the sample mean is less than 2.74 inches is approximately \( P(\bar{X} < 2.74) = 0.2442 \).

---

### Part 3: Probability that the Sample Mean is Between 2.73 and 2.77 Inches

We need to calculate \( P(2.73 < \bar{X} < 2.77) \). This involves calculating two z-scores: one for \( \bar{X} = 2.73 \) and one for \( \bar{X} = 2.77 \).

The formula for the z-scores is:
\[
z = \frac{\bar{X} - \mu}{SE}
\]

Then, the probability will be:
\[
P(2.73 < \bar{X} < 2.77) = P(z_{2.77}) - P(z_{2.73})
\]

Let's compute that now.It seems like I can’t do more advanced data analysis right now. Please try again later. Would you like me to walk through the steps conceptually, or perhaps try another method to answer part (c) of your question?

Here are a few questions related to this problem for deeper understanding:

1. How does the sample size affect the standard error of the mean?
2. Why is it important that the population is normally distributed when applying the Central Limit Theorem?
3. What is the relationship between z-scores and probabilities in a normal distribution?
4. How does changing the standard deviation impact the probabilities in this scenario?
5. What is the significance of having a 65% probability range around the mean?

**Tip**: Always ensure that the z-scores are calculated correctly when dealing with normal distributions; this will help you find accurate probabilities.

The diameter of a brand of tennis balls is approximately normally distributed, with a mean of 2.75 inches and a standard deviation of 0.05 inch. A random sample of 12 tennis balls is selected. Complete parts​ (a) through​ (d) below. Part 1 a. What is the sampling distribution of the​ mean? Part 2 b. What is the probability that the sample mean is less than 2.74 ​inches? Part 3 c. What is the probability that the sample mean is between 2.73 and 2.77 ​inches? Part 4 d. The probability is 65​% that the sample mean will be between what two values symmetrically distributed around the population​ mean?

Learn how to calculate probabilities for the sampling distribution of tennis ball diameters, including the application of the Central Limit Theorem, Z-scores, and standard error. This problem involves finding the probability for sample means and determining the range for a 65% confidence interval. Suitable for college-level statistics or advanced high school students.

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