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Sagot :
Certainly, let's go through the calculation step-by-step to find the expected mean and standard deviation of the sampling distribution of the sample mean when a sample of 40 teams is selected.
### Step 1: Understand the given information
We know the following:
- Population mean ([tex]\(\mu\)[/tex]): \[tex]$750 - Population standard deviation (\(\sigma\)): \$[/tex]185
- Sample size ([tex]\(n\)[/tex]): 40 teams
### Step 2: Calculate the expected mean of the sampling distribution ([tex]\(\mu_{\bar{x}}\)[/tex])
The mean of the sampling distribution of the sample mean is the same as the population mean. Therefore:
[tex]\[ \mu_{\bar{x}} = \mu = \$750 \][/tex]
### Step 3: Calculate the standard deviation of the sampling distribution ([tex]\(\sigma_{\bar{x}}\)[/tex])
The standard deviation of the sampling distribution of the sample mean (often termed the standard error) can be calculated using the formula:
[tex]\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \][/tex]
Plugging in the given values:
[tex]\[ \sigma_{\bar{x}} = \frac{185}{\sqrt{40}} \][/tex]
### Step 4: Perform the division
First, calculate [tex]\(\sqrt{40}\)[/tex]:
[tex]\[ \sqrt{40} \approx 6.3246 \][/tex]
Then divide the population standard deviation by this value:
[tex]\[ \sigma_{\bar{x}} \approx \frac{185}{6.3246} \approx 29.25 \][/tex]
### Step 5: Compile the results
The expected mean of the sampling distribution ([tex]\(\mu_{\bar{x}}\)[/tex]) is:
[tex]\[ \$750 \][/tex]
The standard deviation of the sampling distribution ([tex]\(\sigma_{\bar{x}}\)[/tex]) is:
[tex]\[ \$29.25 \][/tex]
### Step 6: Answer choices
Given the question, select the correct choices:
- [tex]$\sigma_{\bar{x}}=\$[/tex] 185[tex]$ (Incorrect, this is the population standard deviation) - $[/tex]\sigma_{\bar{x}}=\[tex]$ 29.25$[/tex] (Correct)
- [tex]$\sigma_{\overline{\text { x }}}=\$[/tex] 4.63[tex]$ (Incorrect, this value is too small) - $[/tex]\mu_{\bar{x}}=\[tex]$ 118.59$[/tex] (Incorrect, this is not relevant)
- [tex]$\mu_{\bar{x}}=\$[/tex] 18.75[tex]$ (Incorrect, this is not relevant) Therefore, the correct selections are: - $[/tex]\sigma_{\bar{x}}=\[tex]$ 29.25$[/tex]
- [tex]$\mu_{\bar{x}}=\$[/tex] 750$
These reflect the expected mean and standard deviation of the sampling distribution given the sample size and the population parameters.
### Step 1: Understand the given information
We know the following:
- Population mean ([tex]\(\mu\)[/tex]): \[tex]$750 - Population standard deviation (\(\sigma\)): \$[/tex]185
- Sample size ([tex]\(n\)[/tex]): 40 teams
### Step 2: Calculate the expected mean of the sampling distribution ([tex]\(\mu_{\bar{x}}\)[/tex])
The mean of the sampling distribution of the sample mean is the same as the population mean. Therefore:
[tex]\[ \mu_{\bar{x}} = \mu = \$750 \][/tex]
### Step 3: Calculate the standard deviation of the sampling distribution ([tex]\(\sigma_{\bar{x}}\)[/tex])
The standard deviation of the sampling distribution of the sample mean (often termed the standard error) can be calculated using the formula:
[tex]\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \][/tex]
Plugging in the given values:
[tex]\[ \sigma_{\bar{x}} = \frac{185}{\sqrt{40}} \][/tex]
### Step 4: Perform the division
First, calculate [tex]\(\sqrt{40}\)[/tex]:
[tex]\[ \sqrt{40} \approx 6.3246 \][/tex]
Then divide the population standard deviation by this value:
[tex]\[ \sigma_{\bar{x}} \approx \frac{185}{6.3246} \approx 29.25 \][/tex]
### Step 5: Compile the results
The expected mean of the sampling distribution ([tex]\(\mu_{\bar{x}}\)[/tex]) is:
[tex]\[ \$750 \][/tex]
The standard deviation of the sampling distribution ([tex]\(\sigma_{\bar{x}}\)[/tex]) is:
[tex]\[ \$29.25 \][/tex]
### Step 6: Answer choices
Given the question, select the correct choices:
- [tex]$\sigma_{\bar{x}}=\$[/tex] 185[tex]$ (Incorrect, this is the population standard deviation) - $[/tex]\sigma_{\bar{x}}=\[tex]$ 29.25$[/tex] (Correct)
- [tex]$\sigma_{\overline{\text { x }}}=\$[/tex] 4.63[tex]$ (Incorrect, this value is too small) - $[/tex]\mu_{\bar{x}}=\[tex]$ 118.59$[/tex] (Incorrect, this is not relevant)
- [tex]$\mu_{\bar{x}}=\$[/tex] 18.75[tex]$ (Incorrect, this is not relevant) Therefore, the correct selections are: - $[/tex]\sigma_{\bar{x}}=\[tex]$ 29.25$[/tex]
- [tex]$\mu_{\bar{x}}=\$[/tex] 750$
These reflect the expected mean and standard deviation of the sampling distribution given the sample size and the population parameters.
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