Solved on Mar 26, 2024

Find the mean $\mu_{\bar{x}}$ and standard deviation $\sigma_{\bar{x}}$ of the sample mean, given population mean $\mu=83$ , population standard deviation $\sigma=24$ , and sample size $n=36$ .

STEP 1

1. $\mu$ represents the population mean.
2. $\sigma$ represents the population standard deviation.
3. $n$ represents the sample size.
4. $\mu_{\bar{x}}$ represents the mean of the sampling distribution of the sample mean.
5. $\sigma_{\bar{x}}$ represents the standard deviation of the sampling distribution of the sample mean, also known as the standard error of the mean.
6. The Central Limit Theorem applies, which states that the sampling distribution of the sample mean will be approximately normally distributed if the sample size is large enough (typically $n \geq 30$ is considered sufficient).

STEP 2

1. Calculate the mean of the sampling distribution of the sample mean ( $\mu_{\bar{x}}$ ).
2. Calculate the standard deviation of the sampling distribution of the sample mean ( $\sigma_{\bar{x}}$ ), also known as the standard error.

STEP 3

Determine $\mu_{\bar{x}}$ , which is equal to the population mean $\mu$ .
$\mu_{\bar{x}} = \mu$

STEP 4

Calculate $\sigma_{\bar{x}}$ , which is the standard error of the mean, using the formula:
$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

STEP 5

Substitute the given values to find $\mu_{\bar{x}}$ .
$\mu_{\bar{x}} = 83$

STEP 6

Substitute the given values to find $\sigma_{\bar{x}}$ .
$\sigma_{\bar{x}} = \frac{24}{\sqrt{36}}$

STEP 7

Calculate the square root of the sample size $n$ .
$\sqrt{36} = 6$

STEP 8

Divide the population standard deviation $\sigma$ by the square root of the sample size to find $\sigma_{\bar{x}}$ .
$\sigma_{\bar{x}} = \frac{24}{6}$

STEP 9

Complete the calculation to find $\sigma_{\bar{x}}$ .
$\sigma_{\bar{x}} = 4$
The values of $\mu_{\bar{x}}$ and $\sigma_{\bar{x}}$ are: $\mu_{\bar{x}} = 83$ $\sigma_{\bar{x}} = 4$

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Find the mean $\mu_{\bar{x}}$ and standard deviation $\sigma_{\bar{x}}$ of the sample mean, given population mean $\mu=83$ , population standard deviation $\sigma=24$ , and sample size $n=36$ .

STEP 1

STEP 2

1. Calculate the mean of the sampling distribution of the sample mean ( $\mu_{\bar{x}}$ ).
2. Calculate the standard deviation of the sampling distribution of the sample mean ( $\sigma_{\bar{x}}$ ), also known as the standard error.

STEP 3

Determine $\mu_{\bar{x}}$ , which is equal to the population mean $\mu$ .
$\mu_{\bar{x}} = \mu$

STEP 4

Calculate $\sigma_{\bar{x}}$ , which is the standard error of the mean, using the formula:
$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

STEP 5

Substitute the given values to find $\mu_{\bar{x}}$ .
$\mu_{\bar{x}} = 83$

STEP 6

Substitute the given values to find $\sigma_{\bar{x}}$ .
$\sigma_{\bar{x}} = \frac{24}{\sqrt{36}}$

STEP 7

Calculate the square root of the sample size $n$ .
$\sqrt{36} = 6$

STEP 8

Divide the population standard deviation $\sigma$ by the square root of the sample size to find $\sigma_{\bar{x}}$ .
$\sigma_{\bar{x}} = \frac{24}{6}$

STEP 9

Complete the calculation to find $\sigma_{\bar{x}}$ .
$\sigma_{\bar{x}} = 4$
The values of $\mu_{\bar{x}}$ and $\sigma_{\bar{x}}$ are: $\mu_{\bar{x}} = 83$ $\sigma_{\bar{x}} = 4$

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Find the mean μxˉ\mu_{\bar{x}}μxˉ​ and standard deviation σxˉ\sigma_{\bar{x}}σxˉ​ of the sample mean, given population mean μ=83\mu=83μ=83, population standard deviation σ=24\sigma=24σ=24, and sample size n=36n=36n=36.

STEP 1

STEP 2

1. Calculate the mean of the sampling distribution of the sample mean (μxˉ\mu_{\bar{x}}μxˉ​).2. Calculate the standard deviation of the sampling distribution of the sample mean (σxˉ\sigma_{\bar{x}}σxˉ​), also known as the standard error.

STEP 3

Determine μxˉ\mu_{\bar{x}}μxˉ​, which is equal to the population mean μ\muμ.μxˉ=μ \mu_{\bar{x}} = \mu μxˉ​=μ

STEP 4

Calculate σxˉ\sigma_{\bar{x}}σxˉ​, which is the standard error of the mean, using the formula:σxˉ=σn \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} σxˉ​=n​σ​

STEP 5

Substitute the given values to find μxˉ\mu_{\bar{x}}μxˉ​.μxˉ=83 \mu_{\bar{x}} = 83 μxˉ​=83

STEP 6

Substitute the given values to find σxˉ\sigma_{\bar{x}}σxˉ​.σxˉ=2436 \sigma_{\bar{x}} = \frac{24}{\sqrt{36}} σxˉ​=36​24​

STEP 7

Calculate the square root of the sample size nnn.36=6 \sqrt{36} = 6 36​=6

STEP 8

Divide the population standard deviation σ\sigmaσ by the square root of the sample size to find σxˉ\sigma_{\bar{x}}σxˉ​.σxˉ=246 \sigma_{\bar{x}} = \frac{24}{6} σxˉ​=624​

STEP 9

Complete the calculation to find σxˉ\sigma_{\bar{x}}σxˉ​.σxˉ=4 \sigma_{\bar{x}} = 4 σxˉ​=4The values of μxˉ\mu_{\bar{x}}μxˉ​ and σxˉ\sigma_{\bar{x}}σxˉ​ are: μxˉ=83 \mu_{\bar{x}} = 83 μxˉ​=83 σxˉ=4 \sigma_{\bar{x}} = 4 σxˉ​=4

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Find the mean $\mu_{\bar{x}}$ and standard deviation $\sigma_{\bar{x}}$ of the sample mean, given population mean $\mu=83$ , population standard deviation $\sigma=24$ , and sample size $n=36$ .

1. Calculate the mean of the sampling distribution of the sample mean ( $\mu_{\bar{x}}$ ).
2. Calculate the standard deviation of the sampling distribution of the sample mean ( $\sigma_{\bar{x}}$ ), also known as the standard error.

Determine $\mu_{\bar{x}}$ , which is equal to the population mean $\mu$ .
$\mu_{\bar{x}} = \mu$

Calculate $\sigma_{\bar{x}}$ , which is the standard error of the mean, using the formula:
$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

Substitute the given values to find $\mu_{\bar{x}}$ .
$\mu_{\bar{x}} = 83$

Substitute the given values to find $\sigma_{\bar{x}}$ .
$\sigma_{\bar{x}} = \frac{24}{\sqrt{36}}$

Calculate the square root of the sample size $n$ .
$\sqrt{36} = 6$

Divide the population standard deviation $\sigma$ by the square root of the sample size to find $\sigma_{\bar{x}}$ .
$\sigma_{\bar{x}} = \frac{24}{6}$

Complete the calculation to find $\sigma_{\bar{x}}$ .
$\sigma_{\bar{x}} = 4$
The values of $\mu_{\bar{x}}$ and $\sigma_{\bar{x}}$ are: $\mu_{\bar{x}} = 83$ $\sigma_{\bar{x}} = 4$