91Ó°ÊÓ

To compute ${\mathrm{σ}}_{\stackrel{¯}{x}}$for samples of sizes $1,2,3,4,$and $5$and compare the answers with Table $7.6$.

Let, Population standard deviation is $\mathrm{σ}=3.41$. And the Population size is $N=5$

To compute ${\mathrm{σ}}_{\stackrel{¯}{x}}$for samples of sizes$1,2,3,4,$ and $5$ and compare the answers with Table $7.6$ and explain why the equation $(7.2)$ generally yields such poor approximations to the true values.

Let, the population standard deviation is $\mathrm{σ}=3.41$.

The appropriate formula for determining the value of ${\mathrm{σ}}_{\overline{x}}$is equation $7.1$, i.e.,${\mathrm{σ}}_{\stackrel{¯}{x}}=\frac{\mathrm{σ}}{\sqrt{n}}×\sqrt{\frac{N−n}{N−1}}$, when using the basic random sampling without replacement technique to pick the samples from the population.
Equation $7.2$is, ${\mathrm{σ}}_{\stackrel{¯}{x}}=\frac{\mathrm{σ}}{\sqrt{n}}$
Because the sample size is small in comparison to the population size, there is little difference between sampling without and with replacement. In other words equations $7.1$and $7.2$produce roughly the same value of ${\mathrm{σ}}_{\overline{X}}$ for small sample sizes.
As a rule of thumb, if the sample size does not exceed $5\%$of the population size (i.e. $n≤0.05N$), the sample size is small relative to the population size. However, in this case, the smallest sample size, $n=1,$is equal to $20\%$of the population size, $N=5$.

This is why the equation $7.2$in this problem approximates the true values of ${\mathrm{σ}}_{\overline{x}}$ so poorly.

To find the percentages of the population size of samples of sizes $1,2,3,4,$ and $5$.

The population size is $N=5$.

Hence, the percentages of the population size of samples of sizes $1,2,3,4,$and $5$are obtained.