**the mean trout lengths based on samples of size $5$.****the average SAT score of a sample of high school students.****the average male height based on samples of size $30$.****the heights of college students at a sampled university****all mean trout lengths in a sampled lake.**

In this question, we need to choose the statements which best describe the sampling distribution.

A population refers to the whole group about which the conclusions are drawn. A sample is a particular group from which the data is collected. The sample size is always less than the population size.

A sampling distribution is a statistic that calculates the likelihood of an event based on data from a small subset of a larger population. It represents the frequency distribution of how far apart various outcomes will be for a particular population and is also called a finite-sample distribution. It relies upon several factors, including the statistic, sample size, sampling process, and overall population. It is used to calculate statistics for a given sample such as mean, range, variance, and standard deviation.

Inferential statistics require sampling distributions because they make it easier to understand a specific sample statistic regarding other possible values.

**Expert Answer**

In this question:

The mean trout lengths based on samples of size $5$,

The average male height based on samples of size $30$,

both are possible sampling distributions because they are samples drawn from a population.

However, in the statements,

Average SAT score of a sample of high school students,

Heights of college students at a sampled university,

All mean trout lengths in a sampled lake,

Average SAT score, heights of college students, and all mean trout lengths are approximated as population.

Hence, mean trout lengths based on samples of size $5$

and average male height based on samples of size $30$ are the correct examples of the sampling distribution.

The sampling distribution of sample proportions is discussed in the following examples to have a better understanding of the sampling distribution.

## Example 1

Assume that $34\%$ of people own a smartphone. If a random sample of $30$ people is taken, find the probability that the proportion of samples who owned smartphones is between $40\%$ and $45\%$.

In this problem we have the following data:

Mean $=\mu_{\hat{p}}=p=0.34$

$n=30$.

Since, $np=(30)(0.34)=10.2$ and $n(1-p)=30(1-0.34)=19.8$ are greater than $5$, so we can say that $\hat{p}$ has the sampling distribution which is approximately normal with mean $\mu=0.34$ and standard deviation:

$\sigma_{\hat{p}}=\sqrt{\dfrac{p(1-p)}{30}}=\sqrt{\dfrac{0.34(1-0.34)}{30}}=0.09$

And so,

$P(0.4<\hat{p}<0.45)$ $=P\left(\dfrac{0.4-0.34}{0.09}<\dfrac{\hat{p}-p}{\sigma_{\hat{p}}}<\dfrac{0.45-0.34}{0.09}\right)$

$\approx P(0.67<Z<1.22)$

$=P(Z<1.22)-P(Z<0.67)$

$=0.3888-0.2486$

$=0.1402$

## Example 2

Consider the data in example 1. If a random sample of $63$ people was surveyed, what is the probability that more than $40\%$ of them own a smartphone?

Since,

$np=63(0.34)=21.42$ and $n(1-p)=63(1-0.34)=41.58$ are greater than $5$, therefore the sampling distribution of sample proportion is approximately normal with mean $\mu=0.34$ and standard deviation:

$\sigma_{\hat{p}}=\sqrt{\dfrac{p(1-p)}{63}}=\sqrt{\dfrac{0.34(1-0.34)}{63}}=0.06$

So, $P(\hat{p}>0.4)=\left(\dfrac{\hat{p}-p}{\sigma_{\hat{p}}}>\dfrac{0.4-0.34}{0.06}\right)$

$\approx P(Z>1)$

$=1-P(Z<1)$

$=1-0.3413$

$=0.6587$