Question

Do social robots walk or roll? Refer to the InternationalConference on Social Robotics(Vol. 6414, 2010) studyof the trend in the design of social robots, Exercise 6.48(p. 357). Recall that you used a 99% confidence interval toestimate the proportion of all social robots designed withlegs, but no wheels. How many social robots would need tobe sampled in order to estimate the proportion to within.075 of its true value?

Short answer

286 social robots would need to be sampled in order to estimate the proportion to within.075 of its true value

Step-by-step solution

Given information

Referring to Exercise 6.48(p. 357).

Finding the sample size

Here the sample proportion$\widehat{p}=\frac{63}{106}=0.59$

$\begin{array}{c}\widehat{q}=1-\widehat{p}\\ =1-0.59434\\ =0.41\end{array}$

Here the standard error is 0.075

The critical value for a 99% confidence interval is$z_{\alpha /2}=z_{0.01/2}=z_{0.005}=2.576$

$\begin{array}{c}SE=z_{\alpha /2}\sqrt{\frac{\widehat{p}\widehat{q}}{n}}\\ n={z^2}_{\alpha /2}\frac{\widehat{p}\widehat{q}}{S{E}^2}\\ n={2.576}^2\times \frac{0.59\times 0.41}{{0.075}^2}\\ n=285.37\\ n\approx 286\end{array}$

The required sample size is 286

286 social robots would need to be sampled in order to estimate the proportion to within.075 of its true value