91Ó°ÊÓ

The proportion of all social robots designed with legs but no wheels is $p=0.40$.

A random sample of size $n=106$is selected.

Let $\hat{p}$represents the sample proportion of social robots designed with legs but no wheels.

a.

The mean of the sampling distribution of $\hat{p}$is obtained as:

$\begin{array}{rcl}E\left(\hat{p}\right)& =& p\\ & =& 0.40\end{array}$.

Since$\mathit{E}\mathbf{\left(}\hat{\mathbf{p}}\mathbf{\right)}\mathbf{=}\mathit{p}$ .

Therefore, $E\left(\hat{p}\right)=0.40$.

The standard deviation of the sampling distribution of $\hat{p}$is obtained as:

$\begin{array}{rcl}{\mathrm{σ}}_{\hat{p}}& =& \sqrt{\frac{p\left(1-p\right)}{n}}\\ & =& \sqrt{\frac{0.40×0.60}{106}}\\ & =& \sqrt{0.002264}\\ & =& 0.0476\end{array}$.

Therefore, ${\mathrm{σ}}_{\hat{p}}=0.0476$.

b.

Here,

$\begin{array}{rcl}n\hat{p}& =& 106×0.40\\ & =& 42.4\\ & >& 15\end{array}$,

and

$\begin{array}{rcl}dn\left(1-\hat{p}\right)& =& 106×0.60\\ & =& 63.6\\ & >& 15\end{array}$.

Since the conditions to use Central Limit Theorem are satisfied. Thus, the shape of the sampling distribution of $\hat{p}$is approximately normal.

c.

The probability that $\hat{p}$is greater than 0.59 is obtained as:

$\begin{array}{rcl}P\left(\hat{p}>0.59\right)& =& P\left(\frac{\hat{p}-p}{{\mathrm{σ}}_{\hat{p}}}>\frac{0.59-p}{{\mathrm{σ}}_{\hat{p}}}\right)\\ & =& P\left(Z>\frac{0.59-0.40}{0.0476}\right)\\ & =& P\left(Z>\frac{0.19}{0.0476}\right)\\ & =& P\left(Z>3.99\right)\\ & =& 1-P\left(Zâ\copyright ½3.99\right)\\ & =& 1-0.999967\\ & =& 0.000033\end{array}$

.

The probability is obtained by using calculators.

Hence, the required probability is 0.000033.

d.

A researcher found that 63 of the 106 robots were built with legs only. Therefore a sample proportion is:

.$\begin{array}{rcl}\hat{p}& =& \frac{63}{106}\\ & =& 0.59\end{array}$

Since the sample proportion is greater than the population proportion, the result cast doubt on the assumption that 40% of all social robots are designed with legs.

If the sample proportion value was near the population, then there was no doubt about the assumption.