Population data: $1,2,3,4,5,6$

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 $0.5$.

The mean $\mu$is given below,

$$\begin{gathered} \mu =\frac{\sum _{x_i}}{N} \\ =\frac{1+2+3+4+5+6}{6} \\ =\frac{21}{6} \\ =3.5 \end{gathered}$$

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 $n=5$,

If the sample size taken $n=6$

We can observe that from the dot plot there is no dot corresponding to $\mu =3.5$ when n is $1$.

Hence, the probability that sample mean will be equal to population mean$$\begin{gathered} =\frac{0}{6} \\ =0 \end{gathered}$$.

Similarly, the probability that sample mean will be equal to population mean when n is $2$is $\frac{3}{15}=\frac{1}{5}$(As there are $3$dots corresponding $\mu =3.5$)

The probability that sample mean will be equal to population mean when n is $3$is $\frac{0}{20}=0$(As there are $3$dots corresponding $\mu =3.5$)

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 when n is $4$is $\frac{3}{15}=\frac{1}{5}$(As there are $3$dots corresponding $\mu =3.5$)

The probability that sample mean will be equal to population mean when n is $5$is $\frac{0}{6}=0$.

The probability that sample mean will be equal to population mean for $n=6$is $1$.

Number of dots within $0.5$or less of role="math" localid="1652596657200" $\mu =3.5$is $2$out of $6$ when n is $1$.

Hence, the probability that $\overline{x}$will be within $0.5$or less of $\mu$is $\frac{2}{6}=\frac{1}{3}$.

Number of dots within $0.5$or less of role="math" localid="1652596659615" $\mu =3.5$is $7$out of $15$ when n is $2$.

Hence, the probability that $\overline{x}$will be within $0.5$or less of $\mu$is $\frac{7}{15}$.

Number of dots within $0.5$or less of role="math" localid="1652596662174" $\mu =3.5$is $12$out of $20$ for n is $3$.

Hence, the probability that $\overline{x}$will be within $0.5$or less of $\mu$is $\frac{12}{20}=\frac{3}{5}$.

Number of dots within $0.5$or less of role="math" localid="1652596648815" $\mu =3.5$is $11$out of $15$ when nis $4$.

Hence, the probability that $\overline{x}$will be within $0.5$or less of $\mu$is $\frac{11}{15}$.

Number of dots within data-custom-editor="chemistry" $0.5$or less of role="math" localid="1652596728978" $\mu =3.5$is $6$out of $6$ when n is data-custom-editor="chemistry" $5$.

Hence, the probability that $\overline{x}$will be within $0.5$or less of $\mu$is $\frac{6}{6}=1$.

Number of dots within $0.5$or less of $\mu =3.5$is $1$out of $1$ when n is $6$.

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