Question

Consider sampling heights from the population of all female college soccer players in the United States. Assume the mean height of female college soccer players in the United States is LaTeX: \mu=66μ = 66 inches and the standard deviation is LaTeX: \sigma=3.5σ = 3.5 inches. Suppose we randomly sample 100 values from this population and compute the mean, then repeat this sampling process 5,000 times and record all the means we get.
Which of the following is the best approximation for the standard deviation of the 5,000 sample means?

Group of answer choices

0.035

0.35

3.5

Answer 1

Answer:

The approximated value of the standard deviation is 0.35.

Step-by-step explanation:

According to the Central Limit Theorem if we have an unknown population with mean μ and standard deviation σ and appropriately huge random samples (n > 30) are selected from the population with replacement, then the distribution of the sample mean will be approximately normally distributed.  

Then, the mean of the distribution of sample means is given by,

$\mu_{\bar x}=\mu$

And the standard deviation of the distribution of sample means is given by,

$\sigma_{\bar x}=\frac{\sigma}{\sqrt{n}}$

The information provided is:

n = 100

σ = 3.5

μ = 66

As the sample size is quite large, i.e. n = 100 > 30, the central limit theorem can be applied to approximate the sampling distribution of sample mean by the Normal distribution.

Then the approximated value of the standard deviation of sampling distribution of sample mean is:

$\sigma_{\bar x}=\frac{\sigma}{\sqrt{n}}$

    $=\frac{3.5}{\sqrt{100}}\\\\=0.35$

Thus, the approximated value of the standard deviation is 0.35.