Questions: The heights of fully grown trees of a specific species are normally distributed, with a mean of 65.5 feet and a standard deviation of 5.00 feet. Random samples of size 12 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 μ𝑥̄= The standard error of the sampling distribution is σ𝑥̄= (Round to two decimal places as needed.) Choose the correct graph of the sampling distribution below. A. B. C.

The heights of fully grown trees of a specific species are normally distributed, with a mean of 65.5 feet and a standard deviation of 5.00 feet. Random samples of size 12 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 μ𝑥̄=
The standard error of the sampling distribution is σ𝑥̄=
(Round to two decimal places as needed.)
Choose the correct graph of the sampling distribution below.
A.
B.
C.

Transcript text: The heights of fully grown trees of a specific species are normally distributed, with a mean of 65.5 feet and a standard deviation of 5.00 feet. Random samples of size 12 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 $\mu_{\bar{x}}=\square$ $\square$ The standard error of the sampling distribution is $\sigma_{\bar{x}}=$ $\square$ (Round to two decimal places as needed.) Choose the correct graph of the sampling distribution below. A. B. c.

Solution

Solution Steps

Step 1: Identify the mean of the sampling distribution

The mean of the sampling distribution (μx̄) is the same as the mean of the population (μ). Given that the mean of the population is 65.5 feet, the mean of the sampling distribution is: \[ \mu_{x̄} = 65.5 \]

Step 2: Calculate the standard error of the sampling distribution

The standard error of the sampling distribution (σx̄) is calculated using the formula: \[ \sigma_{x̄} = \frac{\sigma}{\sqrt{n}} \] where σ is the standard deviation of the population and n is the sample size. Given that σ = 5.00 feet and n = 12: \[ \sigma_{x̄} = \frac{5.00}{\sqrt{12}} \approx 1.44 \]

Step 3: Choose the correct graph of the sampling distribution

The sampling distribution will be normally distributed with a mean of 65.5 and a standard error of approximately 1.44. The correct graph should reflect these parameters. Among the given options, graph B (with a mean of 65.5 and a spread that matches the calculated standard error) is the correct choice.