Questions: The heights of fully grown trees of a specific species are normally distributed, with a mean of 51.0 feet and a standard deviation of 6.25 feet. Random samples of size 13 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 51.0 feet and a standard deviation of 6.25 feet. Random samples of size 13 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 51.0 feet and a standard deviation of 6.25 feet Random samples of size 13 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 $\mu_{\bar{x}}=$ $\square$ The standard error of the sampling distribution is $\sigma_{\overline{\mathrm{x}}}=$ $\square$ (Round to two decimal places as needed.) Choose the correct graph of the sampling distribution below. A. B. C.

Solution

Solution Steps

Step 1: Calculate the mean of the sampling distribution

The mean of the sampling distribution (μ_x̄) is equal to the mean of the population (μ). The problem states that the mean height of the trees is 51.0 feet. Therefore, the mean of the sampling distribution is also 51.0 feet.

Step 2: Calculate the standard error of the sampling distribution.

The standard error of the sampling distribution (σ_x̄) is calculated using the formula σ_x̄ = σ / √n, where σ is the population standard deviation and n is the sample size. In this case, σ = 6.25 feet and n = 13. So, σ_x̄ = 6.25 / √13 ≈ 1.73. Rounding to two decimal places, the standard error is 1.73 feet.

Step 3: Identify the correct graph.

The correct graph will be a normal distribution centered at the mean of the sampling distribution (51.0 feet) and have a standard deviation reflected by the standard error (1.73 feet). We are looking for a graph centered at 51.0, and the values on the x-axis should increment/decrement roughly by the value of the standard error of the sample mean (1.73). Therefore graph C is the correct graph as it peaks at 51.0 and covers approximately 2 standard deviations away from 51.0 (51.0 +/- 2*1.73 = 51.0 +/- 3.46 = [47.54, 54.46]).