91Ó°ÊÓ

Consider the given question,

The population data is$2,5,8$.

The mean $\mathrm{μ}$is given below,

$$\begin{gathered} \mathrm{μ}=\frac{\underset{x_i}{∑}}{N} \\ =\frac{2+5+8}{3} \\ =\frac{15}{3} \\ =5 \end{gathered}$$

For each of the possible sample sizes, we construct a table.

If the sample size taken $n=1$,

If the sample size taken $n=2$,

If the sample size taken $n=3$,

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

Hence, the probability that sample mean will be equal to population mean for $n=1$is role="math" localid="1652545895359" $\frac{1}{3}$.

Similarly, the probability that sample mean will be equal to population mean for data-custom-editor="chemistry" $\mathrm{n}=2$is $\frac{1}{3}$.

From the dot plot, we can see that there is no dot corresponding to $\mathrm{μ}=5$.

The probability that sample mean will be equal to population mean for$n=3$is$0$.

Number of dots within $0.5$or less of $\mathrm{μ}=5$is $1$out of $3$ when n is $1$and $2$.

For $n=3$, there is no dots are included in the sample mean $\stackrel{¯}{x}$will be $0.5$or less of $\mathrm{μ}$.

Hence, the probability that $\stackrel{¯}{x}$will be within $0.5$or less of $\mathrm{μ}$is for $n=1$is $\frac{1}{3}$.

Similarly, the probability that $\stackrel{¯}{x}$will be within $0.5$or less of $\mathrm{μ}$for $n=2$is $\frac{1}{3}$.

And the probability that $\stackrel{¯}{x}$ will be withindata-custom-editor="chemistry" $0.5$or less of$\mathrm{μ}$for$n=3$is$0$.