91影视

The researchers discovered a random sample of 106 social robots via a Web search as well as discovered that 63 of them had legs but no wheels. One must calculate the following.

a.99% confidence interval for the proportion of all social robots designed with legs but no wheels

b. Validity of the assumption in exercise 5.42

Here the confidence coefficient is 99%, hence$伪=1\%$.Now from the standard normal distribution table,$z_{\frac{伪}{2}}=2.576$

Here sample size is n=106 and x=63, where x is the number of success

The sample proportion is

$\begin{array}{rcl}\hat{p}& =& \frac{x}{n}\\ & =& \frac{63}{106}\\ & =& 05943\\ & & \end{array}$

The margin of error

$\begin{array}{rcl}E& =& z_{\frac{伪}{2}}脳\sqrt{\frac{\hat{p}\left(1-\hat{p}\right)}{n}}\\ & =& 2.576脳\sqrt{\frac{0.5943脳\left(1-0.5943\right)}{106}}\\ & =& 0.1228\\ & & \end{array}$

Hence 99% lower limit:$\hat{p}-E=0.5943-0.1228=0.4715$

99% upper limit:$\hat{p}+E=0.5943+0.1228=0.7171$

The 99% confidence interval is$\left(0.4715,0.7171\right)$

We are 99% confident that the true proportion lies between the interval$\left(0.4715,0.7171\right)$

The 99% confidence interval is$\left(0.4715,0.7171\right)$ .Since 40%=0.4 does not contain in the interval, $\left(0.4715,0.7171\right)$then the assumption is not valid.