Do social robots walk or roll? Refer to the International Conference on Social Robotics (Vol. 6414, 2010) study of the trend in the design of social robots, Exercise 5.44 (p. 320). The researchers obtained a random sample of 106 social robots through a Web search and determined that 63 were designed with legs, but no wheels.

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

b. In Exercise 5.42, you assumed that 40% of all social robots are designed with legs but no wheels. Comment on the validity of this assumption.

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$\alpha =1\%$.Now from the standard normal distribution table,$z_{\frac{\alpha }{2}}=2.576$

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

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

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.