A population of values has a normal distribution with and . A randorn sample of size
is drawn.
a. What is the mean of the distribution of sample means?

b. What is the standard deviation of the distribution of sample means? Round your answer to two
decimal places.

Ask by Kirk Huang. in the United States

Mar 14,2025

Question

A population of values has a normal distribution with and . A randorn sample of size
is drawn.
a. What is the mean of the distribution of sample means?

b. What is the standard deviation of the distribution of sample means? Round your answer to two
decimal places.

Ask by Kirk Huang. in the United States

Mar 14,2025

UpStudy AI · Accepted Answer

To solve the problem, we need to use the properties of the sampling distribution of the sample mean.

### Given:
- Population mean ($$ \mu $$) = 26.1
- Population standard deviation ($$ \sigma $$) = 91.3
- Sample size ($$ n $$) = 130

### a. Mean of the distribution of sample means ($$ \mu_{\bar{x}} $$):
The mean of the distribution of sample means is equal to the population mean. Therefore:

$$
\mu_{\bar{x}} = \mu = 26.1
$$

### b. Standard deviation of the distribution of sample means ($$ \sigma_{\bar{x}} $$):
The standard deviation of the distribution of sample means (also known as the standard error) is calculated using the formula:

$$
\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}
$$

Now, we will calculate $$ \sigma_{\bar{x}} $$:

1. Calculate $$ \sqrt{n} $$:
   $$
   \sqrt{n} = \sqrt{130}
   $$

2. Calculate $$ \sigma_{\bar{x}} $$:
   $$
   \sigma_{\bar{x}} = \frac{91.3}{\sqrt{130}}
   $$

Let's perform the calculations.

First, calculate $$ \sqrt{130} $$ and then use it to find $$ \sigma_{\bar{x}} $$.

Calculating $$ \sqrt{130} $$:

$$
\sqrt{130} \approx 11.4018
$$

Now, substituting this value into the formula for $$ \sigma_{\bar{x}} $$:

$$
\sigma_{\bar{x}} = \frac{91.3}{11.4018} \approx 8.00
$$

### Final Answers:
- a. $$ \mu_{\bar{x}} = 26.1 $$
- b. $$ \sigma_{\bar{x}} \approx 8.00 $$ (rounded to two decimal places)
a. The mean of the distribution of sample means is 26.1. b. The standard deviation of the distribution of sample means is approximately 8.00.

A population of values has a normal distribution with and . A randorn sample of size
is drawn.
a. What is the mean of the distribution of sample means?

b. What is the standard deviation of the distribution of sample means? Round your answer to two
decimal places.

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Latest Statistics Questions

A population of values has a normal distribution with and . A randorn sample of size is drawn. a. What is the mean of the distribution of sample means? b. What is the standard deviation of the distribution of sample means? Round your answer to two decimal places.

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Latest Statistics Questions

A population of values has a normal distribution with and . A randorn sample of size
is drawn.
a. What is the mean of the distribution of sample means?

b. What is the standard deviation of the distribution of sample means? Round your answer to two
decimal places.