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

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 $238$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$1,2,3,4$.

The mean $\mu$is given below,

$$\begin{gathered} \mu =\frac{\sum _{x_i}}{N} \\ =\frac{1+2+3+4}{4} \\ =\frac{10}{4} \\ =2.5 \end{gathered}$$

If the sample size taken $n=2$,

If the sample size taken $n=3$,

If the sample size taken $\mathrm{n}=4$,

We can observe that from the dot plot there is no dot corresponding to $\mu =2.5$ when n is $1$.

Hence, the probability that sample mean will be equal to population mean$$\begin{gathered} =\frac{0}{4} \\ =0 \end{gathered}$$

Similarly, the probability that sample mean will be equal to population mean for $$\begin{gathered} =\frac{2}{6} \\ =\frac{1}{3} \end{gathered}$$(As there are $2$dots corresponding $\mu =2.5$)

The probability that sample mean will equal to population mean for $n=3$is $1$(As there are one dots corresponding $\mu =2.5$)

We need to 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 the probability that $\overline{x}$will be within $0.5$or less of $\mu$.

Number of dots within $0.5$or less of $\mu =2.5$is $2$out of $4$for $n=1$.

Hence, the probability that $\overline{x}$will be within $0.5$or less of $\mu$is $\frac{2}{4}=\frac{1}{2}$.

Number of dots within $0.5$or less of $\mu =2.5$is $4$out of role="math" localid="1652554136075" $6$for $n=2$.

Hence, the probability that $\overline{x}$will be within $0.5$or less of $\mu$is $\frac{4}{4}=1$.

Number of dots within $0.5$or less data-custom-editor="chemistry" $\mathrm{\mu }=2.5$is $4$out of data-custom-editor="chemistry" $4$for $n=3$.

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