91Ó°ÊÓ

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. In a study of Burger King drive-through orders, it was found that 264 orders were accurate and 54 were not accurate. For McDonald's, 329 orders were found to be accurate while 33 orders were not accurate (based on data from \(Q S R\) magazine). Use a 0.05 significance level to test the claim that Burger King and McDonald's have the same accuracy rates. a. Test the claim using a hypothesis test. b. Test the claim by constructing an appropriate confidence interval. c. Relative to accuracy of orders, does either restaurant chain appear to be better?

Step by step solution

01

- Formulate Hypotheses

First, identify the null hypothesis (ewlineewline \(H_0\): Burger King and McDonald's have the same accuracy rate) and the alternative hypothesis (ewlineewline \(H_1\): The accuracy rates are different).

02

- Calculate Sample Proportions

Calculate the sample proportion for each group:ewlineewline For Burger King, \(\hat{p}_1 = \frac{264}{264 + 54} = 0.830\)ewlineewline For McDonald's, \(\hat{p}_2 = \frac{329}{329 + 33} = 0.909\)

03

- Pooled Sample Proportion

Calculate the pooled sample proportion using:ewlineewline \(\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}\)ewlineewline Here, \(x_1 = 264\), \(x_2 = 329\), \(n_1 = 318\), and \(n_2 = 362\):ewlineewline \(\hat{p} = \frac{264 + 329}{318 + 362} = 0.872\)

04

- Test Statistic

Calculate the test statistic using: ewlineewline \(z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}\)ewlineewline Plugging in the values:ewlineewline \(z = \frac{0.830 - 0.909}{\sqrt{0.872(1 - 0.872)(\frac{1}{318} + \frac{1}{362})}} = -2.64\)

05

- P-value

Find the P-value for \(z = -2.64\).ewlineewline For a two-tailed test, the P-value is twice the area to the left of \(z = -2.64\). Using standard normal distribution tables or technology, find that the P-value \(\approx 0.0082\).

06

- Conclusion About the Null Hypothesis

Compare the P-value to the significance level (\(\alpha = 0.05\)):ewlineewline \(0.0082 < 0.05\), so reject the null hypothesis. There is sufficient evidence to conclude that the accuracy rates are different.

07

- Confidence Interval

Construct the confidence interval for the difference in proportions:ewlineewline \((\hat{p}_1 - \hat{p}_2) \pm z_{\alpha/2}\sqrt{\hat{p}_1(1-\hat{p}_1)/n_1 + \hat{p}_2(1-\hat{p}_2)/n_2}\)ewlineewline Calculate the margin of error:ewlineewline \(1.96 \times \sqrt{(0.830 \times 0.170)/318 + (0.909 \times 0.091)/362} = 0.058\)ewlineewline Confidence interval: \(0.830 - 0.909 \pm 0.058\), which is \([-0.137, -0.021]\).

08

- Final Conclusion

Since 0 is not within the confidence interval, it confirms that there is a significant difference in accuracy rates. McDonald's appears to have a higher order accuracy rate compared to Burger King.

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Key Concepts

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

null hypothesis

When conducting a hypothesis test, the first step is to formulate the null hypothesis (H_0). This is a statement that there is no effect or no difference. It serves as the default or starting assumption. For the accuracy rates at Burger King and McDonald's, the null hypothesis states: Burger King and McDonald's have the same accuracy rate . In other words, any observed difference in sample proportions is due to random chance.

alternative hypothesis

Next, we consider the alternative hypothesis (H_1). This is what you aim to support through your test. It's a statement that contradicts the null hypothesis. For our burger chain accuracy example, the alternative hypothesis states: The accuracy rates are different . This means that any observed difference in sample proportions is not due to random chance, but rather indicates a true difference in the accuracy rates of the orders at the two restaurants.

P-value

The P-value tells us the probability of observing our data, or something more extreme, under the null hypothesis. To find this P-value, we calculate the test statistic and then find where it falls in the standard normal distribution. In this case, the P-value is derived from the test statistic value in a standard normal distribution table. If it is < 0.05 (our significance level), we reject the null hypothesis. Here, with a P-value of approximately 0.0082, which is less than 0.05, we have sufficient evidence to conclude that the accuracy rates are different.

test statistic

A key part of hypothesis testing is calculating the test statistic. This helps us determine how far our sample statistic is from the null hypothesis. For comparing two proportions, the test statistic is often a z-score calculated as:
\(z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}} \)
Plugging in our values:
\(z = \frac{0.830 - 0.909}{\sqrt{\hat{p}(1 - \hat{p})(\frac{1}{318} + \frac{1}{362})}} \), we get -2.64. This value indicates the number of standard deviations our observed difference is from the expected difference under the null hypothesis.

confidence interval

A confidence interval provides a range of values within which we can be certain the true difference lies, based on our sample data. For our burger chain example, we calculate the confidence interval for the difference in proportions as follows:
\((\hat{p}_1 - \hat{p}_2) \pm z_{\alpha/2}\sqrt{\hat{p}_1(1-\hat{p}_1)/n_1 + \hat{p}_2(1-\hat{p}_2)/n_2} \)
Given our data:
The confidence interval would be calculated as
[-0.137, -0.021]. Since 0 does not lie within this interval, it confirms a significant difference in the accuracy rates. This supports our finding that McDonald's appears to have a higher order accuracy rate compared to Burger King.