Population data: $1,2,3,4,5$

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 $293$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 $0.5$.

Consider the given question,

The population data is $1,2,3,4,5$.

The mean $\mu$is given below,

role="math" localid="1652556466666" $$\begin{gathered} \mu =\frac{\sum _{x_i}}{N} \\ =\frac{1+2+3+4+5}{5} \\ =\frac{15}{5} \\ =3 \end{gathered}$$

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 $\mu =3$ 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 when n is $2$is$\frac{2}{10}=\frac{1}{5}$(As there are $2$dots corresponding $\mu =3$)

The probability that sample mean will be equal to population mean when nis$3$are $\frac{2}{10}=\frac{1}{5}$(As there are $2$dots corresponding $\mu =3$)

We can observe that from dot plot there is one dot corresponding to $\mu =3$ when nis $4$.

The probability that sample mean will be equal to population mean when nis$4$$=\frac{1}{5}$

The probability that sample mean will be equal to population mean when nis$5$is $1$.

Number of dots within $0.5$or less of $\mu =3$is data-custom-editor="chemistry" $1$out of $5$ when n is $1$.

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

Number of dots within $0.5$or less of $\mu =3$is $6$out of $10$ when n is $2$.

Hence, the probability that $\overline{x}$will be within $0.5$or less of data-custom-editor="chemistry" $\mathrm{\mu }$is $\frac{6}{10}=\frac{3}{5}$.

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

Hence, the probability that $\overline{x}$will be within $0.5$or less of $\mu$is role="math" localid="1652557460776" $\frac{6}{10}=\frac{3}{5}$.

Number of dots within $0.5$or less of $\mu =3$is $5$out of $5$ when n is $4$.

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

Number of dots within $0.5$or less of $\mu =3$is $1$out of $1$for $n=5$.

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