91Ó°ÊÓ

The mean $\mathrm{μ}$is given below,

$$\begin{gathered} \mathrm{μ}=\frac{\underset{x_i}{∑}}{N} \\ =\frac{1+2+3+4+5+6}{6} \\ =\frac{21}{6} \\ =3.5 \end{gathered}$$

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 $\mathrm{μ}=3.5$ 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 $\mathrm{μ}=3.5$)

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 $\mathrm{μ}=3.5$)

We can observe that from the dot plot there is one dot corresponding to $\mathrm{μ}=5$ 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 $\mathrm{μ}=3.5$)

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" $\mathrm{μ}=3.5$is $2$out of $6$ when n is $1$.

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

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

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

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

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

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

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

Number of dots within data-custom-editor="chemistry" $0.5$or less of role="math" localid="1652596728978" $\mathrm{μ}=3.5$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 $\mathrm{μ}$is $\frac{6}{6}=1$.

Number of dots within $0.5$or less of $\mathrm{μ}=3.5$is $1$out of $1$ when n is $6$.

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