Question

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 2.5 (p. 72). The researchers obtained a random sample of 106 social robots through a Web search and determined the number that was designed with legs but no wheels. Let $\hat{\mathbf{p}}$represent the sample proportion of social robots designed with legs but no wheels. Assume that in the population of all social robots, 40% are designed with legs but no wheels.

a. Give the mean and standard deviation of the sampling distribution of $\hat{\mathbf{p}}$.

b. Describe the shape of the sampling distribution of $\hat{\mathbf{p}}$.

c. Find $\mathit{P}\mathbf{(}\hat{\mathbf{p}}\mathbf{>}\mathbf{.}\mathbf{59}\mathbf{)}$.

d. Recall that the researchers found that 63 of the 106 robots were built with legs only. Does this result cast doubt on the assumption that 40% of all social robots are designed with legs but no wheels? Explain.

Short answer

a. The mean and standard deviation of the sampling distribution of $\hat{p}$are 0.40 and 0.0476, respectively.

b. The shape of the sampling distribution of $\hat{p}$is approximately normal.

c. The probability that$\hat{p}$ is greater than 0.59 is 0.000033.

d. 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.

Step-by-step solution

Given information

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.

Finding the mean and standard deviation of the sample proportion

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:

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

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

Determining the shape of the sampling distribution

b.

Here,

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

and

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

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

Computing the required probability

c.

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

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

.

The probability is obtained by using calculators.

Hence, the required probability is 0.000033.

Explanation of the sample proportion

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.