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Chapter 7: Problem 19

In a study of the accuracy of fast food drive-through orders, Burger King had 264 accurate orders and 54 that were not accurate (based on data from \(Q S R\) magazine). a. Construct a \(99 \%\) confidence interval estimate of the percentage of orders that are not accurate. b. Compare the result from part (a) to this \(99 \%\) confidence interval for the percentage of orders that are not accurate at Wendy's: \(6.2 \%

Short Answer

Expert verified
Burger King's 99% confidence interval for inaccurate orders is (11.38%, 22.58%). Wendy's inaccurate orders are between (6.2%, 15.9%). It suggests Burger King has slightly more inaccurate orders.

Step by step solution

01

- Understand the Given Data

There are 264 accurate orders and 54 inaccurate orders at Burger King. The total number of orders is 264 + 54 = 318.
02

- Calculate the Proportion

The proportion of inaccurate orders is calculated as follows: \(\hat{p} = \frac{54}{318} = 0.1698\)}, {
03

- Find the Standard Error

The standard error (SE) of the proportion is calculated using the formula: \[SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.1698(1 - 0.1698)}{318}} \approx 0.0217\]
04

- Determine the Z-Score for 99% Confidence Level

The Z-score for a 99% confidence level is 2.576.
05

- Calculate the Margin of Error

The margin of error (ME) is calculated as follows: \(ME = Z \times SE = 2.576 \times 0.0217 \approx 0.056\)
06

- Construct the Confidence Interval

The confidence interval can be constructed by adding and subtracting the margin of error from the proportion: \(0.1698 - 0.056 < p < 0.1698 + 0.056 \) \(0.1138 < p < 0.2258 \)
07

- Comparison with Wendy's Confidence Interval

The confidence interval for Burger King's inaccurate orders is (11.38%, 22.58%), while Wendy's is (6.2%, 15.9%). Comparing these intervals, Burger King's accuracy level overlaps with Wendy's lower confidence limit but overall it suggests slightly more inaccuracies than Wendy's.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportion Calculation
The first step in calculating a confidence interval is determining the sample proportion. In our exercise, we analyzed Burger King's order accuracy. We have 54 inaccurate orders out of a total of 318 orders. The proportion of inaccurate orders is calculated by dividing the number of inaccurate orders by the total number of orders: \(\hat{p} = \frac{54}{318} \). This gives us \(\hat{p} = 0.1698\), or about 16.98%. This proportion tells us the fraction of the orders that were not accurate, providing a base for further statistical analysis. Understanding proportions is essential since it represents the likelihood of an event occurring within your sample.
Standard Error
Once we have the proportion, we move on to calculate the standard error (SE). The standard error measures the variability or dispersion of the sample proportion. It's calculated using the formula: \[SE = \sqrt{ \frac{\hat{p}(1 - \hat{p})}{n}}\], where \(\) is the sample size. For our data, it becomes: \[SE = \sqrt{\frac{0.1698(1 - 0.1698)}{318}} \approx 0.0217\]. The standard error helps us understand the extent to which our sample proportion likely differs from the true population proportion. Smaller standard errors suggest more reliable sample estimates.
Margin of Error
With the standard error on hand, the next step is to figure out the margin of error (ME). The margin of error indicates the range in which the true population parameter may lie. For a specified confidence level, we multiply the standard error by the Z-score: \[ME = Z \times SE\]. In our scenario with a 99% confidence level, the Z-score is 2.576. Thus, the margin of error calculation is: \[ME = 2.576 \times 0.0217 \approx 0.056\], or 5.6%. The margin of error adds and subtracts around the sample proportion to give a range that likely contains the population proportion. It tells us how much the sample results can vary.
Z-score
To complete the confidence interval calculation, we must use the Z-score. The Z-score corresponds to the desired confidence level. It represents the number of standard errors to go from the mean in a standard normal distribution. For a 99% confidence level, the Z-score is 2.576. The formula for calculating the Z-score is: \[CI = \hat{p} \pm Z \times SE\]. For our Burger King example, the confidence interval is: \[0.1698 \pm 2.576 \times 0.0217\], which simplifies to: \[0.1138 < p < 0.2258\]. This means we are 99% confident that the true proportion of inaccurate orders falls between 11.38% and 22.58%. The Z-score helps in setting this range based on the desired confidence level.

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