"The trade magazine QSR routinely examines fast-food drive-thru service times. Their recent research indicates that the mean time a car spends in a McDonald’s drive-thru is 167.1 seconds. A McDonald's manager in Salt Lake City feels that she has instituted a drive-thru policy that results in lower drive-thru service times. A random sample of 70 cars results in a mean service time of 163.9 seconds, with a standard deviation of 15.3 seconds. Determine whether the policy is effective in reducing drive-thru service times.


b. Because the cost of instituting the policy is quite high, the quality-control researcher at McDonald’s chooses to test the hypothesis using an α = 0.01 level of significance. Why is this a good idea?"

Verified step by step guidance

1. Understand the context: The manager believes the new policy reduces the mean drive-thru time from the known population mean of 167.1 seconds. The sample mean is 163.9 seconds with a sample size of 70 and a sample standard deviation of 15.3 seconds.
2. Identify the hypotheses for the test: The null hypothesis \(H_0\) states that the mean service time is equal to or greater than 167.1 seconds (no reduction), and the alternative hypothesis \(H_a\) states that the mean service time is less than 167.1 seconds (policy reduces time). Formally, \(H_0: \mu \geq 167.1\) and \(H_a: \mu < 167.1\).
3. Recognize that this is a one-tailed t-test because the population standard deviation is unknown and the sample size is relatively large but not infinite. The test statistic is calculated using the formula: \[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \] where \(\bar{x}\) is the sample mean, \(\mu_0\) is the population mean under the null hypothesis, \(s\) is the sample standard deviation, and \(n\) is the sample size.
4. Understand the significance level \(\alpha = 0.01\): This means the researcher is willing to accept only a 1% chance of incorrectly rejecting the null hypothesis (Type I error). A smaller \(\alpha\) makes the test more stringent, requiring stronger evidence to conclude the policy is effective.
5. Explain why using \(\alpha = 0.01\) is a good idea here: Since instituting the policy is costly, it is important to minimize the risk of falsely concluding the policy reduces service times when it actually does not. A lower significance level reduces this risk, ensuring that the decision to adopt the policy is based on strong statistical evidence.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a null hypothesis in favor of an alternative. In this context, the null hypothesis states that the new policy does not reduce service times, while the alternative suggests it does. The test compares sample data against the assumed population mean to determine if observed differences are statistically significant.

Significance Level (α)

The significance level, denoted by α, is the threshold for rejecting the null hypothesis, representing the probability of a Type I error (false positive). Choosing α = 0.01 means the researcher accepts only a 1% chance of incorrectly concluding the policy reduces service times when it does not. A lower α is appropriate when the cost of a wrong decision is high.

Sample Mean and Standard Deviation

The sample mean is the average service time observed in the sample, and the standard deviation measures the variability of these times. These statistics are used to calculate the test statistic, which helps determine if the sample mean significantly differs from the population mean. Larger samples and smaller variability increase the reliability of the test results.