Question

question #8
given a standard deviation of \\( \sigma=107 \\) and a sample size of \\( n = 55 \\), calculate the
standard deviation of the sampling distribution.
\\( 5.317 \\)
\\( 14.428 \\)
\\( 11.821 \\)
\\( 1.945 \\)

question #9
which of the following statements are true?
i. the larger the sample, the smaller the standard deviation of the sampling
distribution.
ii. provided that the population size is significantly greater than the sample size, the
spread of a sampling distribution is about the same no matter what the sample size.
iii. sampling distributions from non - normal populations are approximately normal
provided \\( n \\) is large.
\\( 1 \\) and iii only
i and ii only
ii only
iii only

Explanation:

Question 8

Step1: Recall the formula for the standard deviation of the sampling distribution

The formula for the standard deviation of the sampling distribution (also known as the standard error) is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \(n\) is the sample size.
Given \(\sigma = 107\) and \(n = 55\).

Step2: Substitute the values into the formula

\(\sigma_{\bar{x}}=\frac{107}{\sqrt{55}}\).
First, calculate \(\sqrt{55}\approx7.416\).
Then, \(\frac{107}{7.416}\approx14.428\).

Question 9

Step1: Analyze statement I

The formula for the standard deviation of the sampling distribution is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\). As \(n\) (sample size) increases, \(\sigma_{\bar{x}}\) (standard deviation of the sampling distribution) decreases. So, statement I is true.

Step2: Analyze statement II

The spread of the sampling distribution is given by \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\), which depends on \(n\) (sample size). So, statement II is false.

Step3: Analyze statement III

According to the Central Limit Theorem, if the sample size \(n\) is large (usually \(n\geq30\)), the sampling distribution of the sample mean \(\bar{X}\) is approximately normal, regardless of the shape of the population distribution. So, statement III is true.

Answer:

• Question 8: \(14.428\)
• Question 9: I and III only