Population data: $2,3,5,7,8$

Part (a): Find the mean, $\mu$, of the variable.

Part (b): For each of the possible sample sizes, construct a table similar to Table $7.2$on the page localid="1652592045497" $293$and draw a dotplot for the sampling for the sampling distribution of the sample mean similar to Fig $7.1$on page $293$.

Part (c): Construct a graph similar to Fig $7.3$and interpret your results.

Part (d): For each of the possible sample sizes, find the probability that the sample mean will equal the population mean.

Part (e): For each of the possible sample sizes, find the probability that the sampling error made in estimating the population mean by the sample mean will be $0.5$or less, that is, that the absolute value of the difference between the sample mean and the population mean is at most$0.5$.

The mean $\mu$is given below,

$$\begin{gathered} \mu =\frac{\sum _{x_i}}{N} \\ =\frac{2+3+5+7+8}{5} \\ =\frac{25}{5} \\ =5 \end{gathered}$$

If the sample size taken $n=2$,

If the sample size taken $n=3$,

If the sample size taken $n=4$,

If the sample size taken $n=5$,

We can observe that from the dot plot there is one dot corresponding to $\mu =5$ when n is $1$.

Hence, the probability that sample mean will be equal to population mean$=\frac{1}{5}$.

Similarly, the probability that sample mean will be equal to population mean whennislocalid="1652594432269" $2$is $$\begin{gathered} =\frac{2}{10} \\ =\frac{1}{5} \end{gathered}$$(As there are $2$dots corresponding $\mu =5$)

We can observe that from the dot plot there is one dot corresponding to $\mu =5$ when n is $4$.

The probability that sample mean will be equal to population mean forlocalid="1652593799606" $n=4$is localid="1652593789207" $\frac{1}{5}$.

The probability that sample mean will be equal to population mean for localid="1652593783763" $n=5$is$1$.

Number of dots within $0.5$or less of $\mu =5$is $1$out of $5$ when n is $1$.

Hence, the probability that $\overline{x}$will be within $0.5$or less of $\mu$is role="math" localid="1652593881183" $\frac{1}{5}$.

Number of dots within $0.5$or less of $\mu =5$is $4$out of $10$ when n is $2$.

Hence, the probability that $\overline{x}$will be within $0.5$or less of $\mu$is $\frac{4}{10}=\frac{2}{5}$.

Number of dots within $0.5$or less of $\mu =5$is $6$out of $10$ when n is $3$.

Hence, the probability that $\overline{x}$will be within $0.5$or less of $\mu$is $\frac{6}{10}=\frac{3}{5}$.

Number of dots within $0.5$or less of $\mu =5$is $4$out of $5$ when n is $4$.

Hence, the probability that $\overline{x}$will be within $0.5$or less of $\mu$is $\frac{4}{5}$.

Number of dots within $0.5$or less of $\mu =5$is role="math" localid="1652594094735" $1$out of $1$ when n is $5$.

Hence, the probability that $\overline{x}$will be within $0.5$or less of $\mu$is$\frac{1}{1}=1$.