Math / Data & Statistics

QuestionDetermine $\mu_{\mathrm{x}}^{-}$ and $\sigma_{\mathrm{x}}^{-}$ from the given parameters of the population and sample size. $\mu=53, \sigma=6, n=35$ $\begin{array}{l} \mu_{\bar{x}}=53 \\ \sigma_{\bar{x}}=\square \end{array}$ (Round to three decimal places as needed.)

Studdy Solution

STEP 1

1. The population mean $\mu$ is given as 53.
2. The population standard deviation $\sigma$ is given as 6.
3. The sample size $n$ is 35.
4. We are assuming a simple random sample from a normally distributed population or a large enough sample size for the Central Limit Theorem to apply.

STEP 2

1. Identify the sample mean $\mu_{\bar{x}}$ .
2. Calculate the standard error of the mean $\sigma_{\bar{x}}$ .

STEP 3

The sample mean $\mu_{\bar{x}}$ is equal to the population mean $\mu$ .
$\mu_{\bar{x}} = \mu = 53$

STEP 4

Calculate the standard error of the mean $\sigma_{\bar{x}}$ using the formula:
$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$
Substitute the given values:
$\sigma_{\bar{x}} = \frac{6}{\sqrt{35}}$

STEP 5

Compute $\sqrt{35}$ and then divide 6 by this value to find $\sigma_{\bar{x}}$ .
$\sqrt{35} \approx 5.916$
$\sigma_{\bar{x}} = \frac{6}{5.916} \approx 1.014$
Round to three decimal places:
$\sigma_{\bar{x}} \approx 1.014$
The standard error of the mean is:
$\boxed{1.014}$

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