Question

Constructing and Interpreting Confidence Intervals. In Exercises 13–16, use the given sample data and confidence level. In each case, (a) find the best point estimate of the population proportion p; (b) identify the value of the margin of error E; (c) construct the confidence interval; (d) write a statement that correctly interprets the confidence interval.

In a study of the accuracy of fast food drive-through orders, McDonald’s had 33 orders that were not accurate among 362 orders observed (based on data from QSR magazine).

Construct a 95% confidence interval for the proportion of orders that are not accurate.

Short answer

(a)Thebest point estimate of the proportion of inaccurate ordersis equal to 0.091.

(b)The margin of error is equal to 0.0297.

(c)The 95% confidence interval estimate of the population proportion of inaccurate orders is equal to (0.0613, 0.1207).

(d)There is 95% confidence that the true proportion of inaccurate orders will lie between the values 0.0613 and 0.1207.

Step-by-step solution

Given information

A sample of McDonald’s orders was collected to check the accuracy of the orders. The sample size is 362. In the sample, 33 orders were inaccurate.

Compute the sample proportion

(a)

The best point estimate of the proportion of inaccurate orders is computed below:

$$\begin{gathered} \hat{p}=\frac{33}{362} \\ =0.091 \end{gathered}$$

Thus, the sample proportion of inaccurate orders equal to 0.091 is the best point estimate of the proportion of inaccurate orders.

Compute the margin of error

(b)

The confidence level is equal to 95%. Thus, the corresponding level of significance is equal to 0.05.

From the standard normal distribution table, the right-tailed value of $z_{\frac{\alpha }{2}}$ for is equal to 1.96.

The margin of error is calculated below:

$$\begin{gathered} E=1.96\times \sqrt{\frac{0.091\times 0.909}{362}} \\ =0.0297 \end{gathered}$$

Thus, the margin of error is equal to 0.0297.

Compute the confidence interval

(c)

The formula for computing the confidence interval estimate of the population proportion is written below:

$CI=\left(\hat{p}-E,\hat{p}+E\right)$

The 95% confidence interval becomes equal to:

$$\begin{gathered} CI=\left(0.091-0.0297,0.09+0.0297\right) \\ =\left(0.0613,0.1207\right) \end{gathered}$$

Therefore, the 95% confidence interval estimate of the population’s proportion of inaccurate orders is equal to (0.0613,0.1207).

Interpretation of the confidence interval

(d)

There is 95% confidence that the true proportion of inaccurate orders will lie between the values 0.0613 and 0.1207.