A measurement is normally distributed with mean=18 and slope=6.4 

. Round answers below to three decimal places.

(a) The mean of the sampling distribution of  for samples of size 11 is: 

(b) The standard deviation of the sampling distribution of  for samples of size 11 is: 

To solve this, let's go over each part. Here’s the approach for each:

Given:

- Mean (\( \mu \)) = 18
- Standard deviation (\( \sigma \)) = 6.4
- Sample size (\( n \)) = 11

Since this is a normal distribution, for any sample of size \( n \), the sampling distribution properties are as follows:

1. **Mean of the sampling distribution of \( \bar{X} \)**:

   \[
   \mu_{\bar{X}} = \mu
   \]

2. **Standard deviation of the sampling distribution of \( \bar{X} \)** (also called the standard error):

   \[
   \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}
   \]

Now let’s apply these formulas to find each answer:

### (a) Mean of the Sampling Distribution
Since the mean of the sampling distribution \( \mu_{\bar{X}} \) is the same as the population mean:

\[
\mu_{\bar{X}} = 18
\]

### (b) Standard Deviation of the Sampling Distribution
The standard deviation of the sampling distribution \( \sigma_{\bar{X}} \) is:

\[
\sigma_{\bar{X}} = \frac{6.4}{\sqrt{11}}
\]

Calculating \( \sigma_{\bar{X}} \):

\[
\sigma_{\bar{X}} = \frac{6.4}{\sqrt{11}} \approx 1.929
\]

### Final Answers:

- **(a)** The mean of the sampling distribution is **18.000**.
- **(b)** The standard deviation of the sampling distribution is **1.929**.

Would you like additional clarification on any part of the solution?

#### Related Questions:

1. How does sample size affect the standard deviation of the sampling distribution?
2. Why does the mean of the sampling distribution equal the population mean?
3. What would happen to the standard deviation if the sample size were doubled?
4. How would the sampling distribution differ for a non-normal population?
5. Can the central limit theorem be applied here with a sample size of 11?

#### Tip:
When calculating the standard deviation of a sampling distribution, remember that increasing the sample size reduces the standard error, making the sample mean more precise.

A measurement is normally distributed with mean=18 and slope=6.4. Round answers below to three decimal places.
(a) The mean of the sampling distribution for samples of size 11 is: 
(b) The standard deviation of the sampling distribution for samples of size 11 is:

Learn how to find the mean and standard deviation of a sampling distribution for normally distributed data. Given a population mean of 18 and a standard deviation of 6.4, this problem demonstrates the application of statistical formulas for a sample size of 11. Ideal for high school students studying statistics.

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