Question

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

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

Given information

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

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.

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.

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.

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.

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.