Math / Data & Statistics

QuestionThe heights of fully grown trees of a specific species are normally distributed, with a mean of 52.5 feet and a standard deviation of 5.75 feet. Random samples of size 17 are drawn from the population. Use the central limit theorem to find the mean and standard error of the sampling distribution. Then sketch a graph of the sampling distribution.
The mean of the sampling distribution is $\mu_{\bar{x}}=$ $\square$ 7.
The standard error of the sampling distribution is $\sigma_{\bar{x}}=$ $\square$ (Round to two decimal places as needed.)

Studdy Solution

STEP 1

1. The population of tree heights is normally distributed.
2. The mean of the population is $\mu = 52.5$ feet.
3. The standard deviation of the population is $\sigma = 5.75$ feet.
4. The sample size is $n = 17$ .
5. The Central Limit Theorem applies, allowing us to use the normal distribution for the sampling distribution of the sample mean.

STEP 2

1. Determine the mean of the sampling distribution.
2. Calculate the standard error of the sampling distribution.
3. Sketch a graph of the sampling distribution.

STEP 3

The mean of the sampling distribution ( $\mu_{\bar{x}}$ ) is equal to the mean of the population ( $\mu$ ).
$\mu_{\bar{x}} = \mu = 52.5$

STEP 4

Calculate the standard error of the sampling distribution ( $\sigma_{\bar{x}}$ ) using the formula:
$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$
Substitute the known values:
$\sigma_{\bar{x}} = \frac{5.75}{\sqrt{17}}$
Calculate $\sqrt{17}$ :
$\sqrt{17} \approx 4.123$
Now calculate $\sigma_{\bar{x}}$ :
$\sigma_{\bar{x}} \approx \frac{5.75}{4.123} \approx 1.394$
Round to two decimal places:
$\sigma_{\bar{x}} \approx 1.39$

STEP 5

To sketch a graph of the sampling distribution, plot a normal distribution curve with:
- Mean ( $\mu_{\bar{x}}$ ) at 52.5 - Standard error ( $\sigma_{\bar{x}}$ ) at 1.39
The graph should be a bell-shaped curve centered at 52.5 with a spread determined by the standard error of 1.39.
The mean of the sampling distribution is:
$\mu_{\bar{x}} = 52.5$
The standard error of the sampling distribution is:
$\sigma_{\bar{x}} = 1.39$

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