# Sample Homework Solution

Sample Question 1: Define the sampling distribution of the mean.

Solution: The sampling distribution of the mean is the distribution of all possible sample means that can be obtained from samples from a certain population of a given size N.

Sample Question 2: Identify three main features of the mean sampling distribution.

Solution: First, the mean of the distribution of sampling mean is the same mean as the population mean of the population from which the samples are obtained. Second, if $$\sigma$$ is the population standard deviation of the population, the standard deviation of the distribution of sample means is $$\frac{\sigma }{\sqrt{n}}$$. Third, due to the Central Limit Theorem, the distribution of sample means is at least approximately normally distributed when the sample size is large enough.

Sample Question 3: The sampling distribution of the mean for a random sample of 35 subjects from a population will have the following properties (true or false).

(a) Shape would approximate a normal curve.

(b) Mean would equal the one sample mean.

(c) Shape would approximate the shape of the population.

(d) Compared to the population variability, the variability would be reduced by a factor equal to the square root of 35.

(e) Mean would equal the population mean.

(f) Variability would equal the population variability.

Solution: (a) TRUE

(b) FALSE

(c) FALSE

(d) TRUE

(e) TRUE

(f) FALSE

Sample Question 4: How high would the population standard deviation have to be for the standard error to be 36 for a sample size of 36?

(a) 1

(b) 2

(c) 5

(d) 100

Solution: (a) We need to solve:

$\frac{\sigma }{\sqrt{36}}=1\,\,\Rightarrow \,\,\,\,\sigma =\sqrt{36}=6$

so then $$\sigma = 6$$.

(b) We need to solve:

$\frac{\sigma }{\sqrt{36}}=2\,\,\Rightarrow \,\,\,\,\sigma =2\times \sqrt{36}=12$

so then $$\sigma = 12$$.

(c) We need to solve:

$\frac{\sigma }{\sqrt{36}}=5\,\,\Rightarrow \,\,\,\,\sigma =5\times \sqrt{36}=30$

so then $$\sigma = 30$$.

(d) We need to solve:

$\frac{\sigma }{\sqrt{36}}=100\,\,\Rightarrow \,\,\,\,\sigma =10\times \sqrt{36}=600$

so then $$\sigma = 12$$.