Testing Claims About Proportions. In Exercises 9–32, 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. Use the P-value method unless your instructor specifies otherwise. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section.

Mickey D’s 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). Use a 0.05 significance level to test the claim that the rate of inaccurate orders is equal to 10%. Does the accuracy rate appear to be acceptable?

Short Answer

Expert verified

Nullhypothesis: The proportion of inaccurate orders is equal to 10%.

Alternativehypothesis: The proportion of inaccurate orders is not equal to 10%.

Teststatistic: -0.56

Criticalvalue: 1.96

P-Value: 0.5754

The null hypothesis is failed to reject.

There is not enough evidence to reject the claim that the proportion of inaccurate orders is equal to 0.10.

The accuracy rate appears to be unacceptable as the percentage of inaccurate orders is 10%.

Step by step solution

01

Given information

Among 362 McDonald’s orders,33 orders were not accurate.

02

Hypotheses

The null hypothesis is written as follows:

The proportion of inaccurate orders is equal to 10%.

\({H_0}:p = 0.10\)

The alternative hypothesis is written as follows:

The proportion of inaccurate orders is not equal to 10%.

\({H_1}:p \ne 0.10\)

The test is two-tailed.

03

Step3:Sample proportion, population proportion, and sample size

The sample proportion of inaccurate orders is equal tothe following:

\[\begin{array}{c}\hat p = \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{inaccurate}}\;{\rm{orders}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{orders}}}}\\ = \frac{{33}}{{362}}\\ = 0.091\end{array}\]

The population proportion of inaccurate orders is equal to p=0.10.

The sample size (n) is equal to 362.

04

Test statistic

The value of the test statistic is computed below:

\(\begin{array}{c}z = \frac{{\hat p - p}}{{\sqrt {\frac{{pq}}{n}} }}\\ = \frac{{0.0912 - 0.10}}{{\sqrt {\frac{{0.10\left( {1 - 0.10} \right)}}{{362}}} }}\\ = - 0.56\end{array}\)

Thus, z=-0.56.

05

Critical value and p-value

Referring to the standard normal distribution table, the critical value of z at \(\alpha = 0.05\) for a two-tailed test is equal to 1.96.

Referring to the standard normal distribution table, the p-value for the test statistic value of -0.56 is equal to 0.5754.

Since the p-value is greater than 0.05, the null hypothesis is failed to reject.

06

Conclusion of the test

There is not enough evidence to reject the claim that the proportion of inaccurate orders is equal to 0.10.

As the percentage of inaccurate orders is 10%, the accuracy rate appears unacceptable, and McDonald’s should lower the rate.

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Most popular questions from this chapter

Test Statistics. In Exercises 13–16, refer to the exercise identified and find the value of the test statistic. (Refer to Table 8-2 on page 362 to select the correct expression for evaluating the test statistic.)

Exercise 7 “Pulse Rates”

Testing Claims About Proportions. In Exercises 9–32, 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. Use the P-value method unless your instructor specifies otherwise. Use the normal distribution as an approximation to the binomial distribution, as described in Part 1 of this section.

Medical Malpractice In a study of 1228 randomly selected medical malpractice lawsuits, it was found that 856 of them were dropped or dismissed (based on data from the Physicians Insurers Association of America). Use a 0.01 significance level to test the claim that most medical malpractice lawsuits are dropped or dismissed. Should this be comforting to physicians?

Critical Values. In Exercises 21–24, refer to the information in the given exercise and do the following.

a. Find the critical value(s).

b. Using a significance level of = 0.05, should we reject H0or should we fail to reject H0?

Exercise 18

Testing Hypotheses. In Exercises 13–24, assume that a simple random sample has been selected and test the given claim. Unless specified by your instructor, use either the P-value method or the critical value method for testing hypotheses. Identify the null and alternative hypotheses, test statistic, P-value (or range of P-values), or critical value(s), and state the final conclusion that addresses the original claim.

Got a Minute? Students of the author estimated the length of one minute without reference to a watch or clock, and the times (seconds) are listed below. Use a 0.05 significance level to test the claim that these times are from a population with a mean equal to 60 seconds. Does it appear that students are reasonably good at estimating one minute?

69 81 39 65 42 21 60 63 66 48 64 70 96 91 65

Final Conclusions. In Exercises 25–28, use a significance level of = 0.05 and use the given information for the following:

a. State a conclusion about the null hypothesis. (Reject H0 or fail to reject H0.)

b. Without using technical terms or symbols, state a final conclusion that addresses the original claim.

Original claim: Fewer than 90% of adults have a cell phone. The hypothesis test results in a P-value of 0.0003.

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