Question

The heights of fully grown trees of a specitic species are nocmally distributed, with a mean of 51.0 teet and a standard deviation of 5.50 feet: Randcm samples of size 12 are drawn from the population. Use the central lifnit theorem to find the moan and standard ocror of the sampling isitribution. Then sketch a graph of the sampling distribution. The mean of the sampling distribution is μ x ​ =51.0 The standard error of the sampling distribution is σ x ​ = (Round to two decimal places as needed.) Choose the correct graph of thes sampling distribution below. A. B

Answer 1

The mean
`(\( \mu_{\bar{x}} \))` of the sampling distribution is 51.0 feet, and the standard error
`(\( \sigma_{\bar{x}} \))` is approximately 1.59 feet.

Explanation:

To find the mean
`(\( \mu_{\bar{x}} \))` and standard error
`(\( \sigma_{\bar{x}} \))` of the sampling distribution, we can use the Central Limit Theorem (CLT). The mean of the sampling distribution
`(\( \mu_{\bar{x}} \))` is equal to the mean of the population
`(\( \mu \)),` which is 51.0 feet.

The standard error of the sampling distribution
`(\( \sigma_{\bar{x}} \))` is calculated using the formula
`\( (\sigma)/(√(n)) \),` where
`\( \sigma \)` is the standard deviation of the population and
`\( n \)` is the sample size. Plugging in the values, we get
`\( \sigma_{\bar{x}} = (5.50)/(√(12)) \approx 1.59 \) feet.`

Now, for the sketch of the sampling distribution, it would typically resemble a normal distribution curve centered around the mean
`(\( \mu_{\bar{x}} \))` with a spread determined by the standard error
`(\( \sigma_{\bar{x}} \))`.

The graph would show that as the sample size increases, the distribution becomes more symmetric and bell-shaped due to the Central Limit Theorem. The x-axis represents the sample means, and the y-axis represents the frequency or probability of each sample mean.

In summary, the mean of the sampling distribution is 51.0 feet, and the standard error is approximately 1.59 feet. The graph of the sampling distribution would exhibit the characteristics of a normal distribution, illustrating the impact of the Central Limit Theorem on the distribution of sample means.