Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Discover detailed solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
Sure, let's solve the problem step-by-step:
Given information:
1. The population mean [tex]\(\mu = 84.1\)[/tex]
2. The population standard deviation [tex]\(\sigma = 42.7\)[/tex]
3. The sample size [tex]\(n = 18\)[/tex]
### Part (a)
What is the mean of the distribution of sample means?
The mean of the distribution of sample means, also called the expected value of the sample mean ([tex]\(\mu_{\bar{x}}\)[/tex]), is equal to the population mean ([tex]\(\mu\)[/tex]). This is a fundamental property of the sampling distribution of the sample mean.
So,
[tex]\[ \mu_{\bar{x}} = \mu = 84.1 \][/tex]
Therefore, the mean of the distribution of sample means is [tex]\(84.1\)[/tex].
### Part (b)
What is the standard deviation of the distribution of sample means?
The standard deviation of the distribution of sample means, also known as the standard error of the mean ([tex]\(\sigma_{\bar{x}}\)[/tex]), is calculated using the formula:
[tex]\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \][/tex]
where [tex]\(\sigma\)[/tex] is the population standard deviation, and [tex]\(n\)[/tex] is the sample size.
Plugging in the given values:
[tex]\[ \sigma_{\bar{x}} = \frac{42.7}{\sqrt{18}} \][/tex]
After computing the above expression, we get:
[tex]\[ \sigma_{\bar{x}} \approx 10.06 \][/tex]
Therefore, the standard deviation of the distribution of sample means, rounded to two decimal places, is [tex]\(10.06\)[/tex].
So, summarizing:
a. The mean of the distribution of sample means is [tex]\(84.1\)[/tex].
b. The standard deviation of the distribution of sample means is [tex]\(10.06\)[/tex].
Given information:
1. The population mean [tex]\(\mu = 84.1\)[/tex]
2. The population standard deviation [tex]\(\sigma = 42.7\)[/tex]
3. The sample size [tex]\(n = 18\)[/tex]
### Part (a)
What is the mean of the distribution of sample means?
The mean of the distribution of sample means, also called the expected value of the sample mean ([tex]\(\mu_{\bar{x}}\)[/tex]), is equal to the population mean ([tex]\(\mu\)[/tex]). This is a fundamental property of the sampling distribution of the sample mean.
So,
[tex]\[ \mu_{\bar{x}} = \mu = 84.1 \][/tex]
Therefore, the mean of the distribution of sample means is [tex]\(84.1\)[/tex].
### Part (b)
What is the standard deviation of the distribution of sample means?
The standard deviation of the distribution of sample means, also known as the standard error of the mean ([tex]\(\sigma_{\bar{x}}\)[/tex]), is calculated using the formula:
[tex]\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \][/tex]
where [tex]\(\sigma\)[/tex] is the population standard deviation, and [tex]\(n\)[/tex] is the sample size.
Plugging in the given values:
[tex]\[ \sigma_{\bar{x}} = \frac{42.7}{\sqrt{18}} \][/tex]
After computing the above expression, we get:
[tex]\[ \sigma_{\bar{x}} \approx 10.06 \][/tex]
Therefore, the standard deviation of the distribution of sample means, rounded to two decimal places, is [tex]\(10.06\)[/tex].
So, summarizing:
a. The mean of the distribution of sample means is [tex]\(84.1\)[/tex].
b. The standard deviation of the distribution of sample means is [tex]\(10.06\)[/tex].
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.