Mastery Check Solutions

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Danell’s Delivery Service claims its average delivery time is 3.0 days. A competitor challenges this claim and states, “It is simply not possible in this region.”

The Consumer Advocacy Group (CAG) opted to conduct an investigation using a random sample of 200 customer deliveries. For the standard deviation, CAG referenced national delivery service data, which indicated a value of 1.5 days.

What is the maximum error of the estimate for a 90% confidence level?

$\begin{matrix} z = 1.645 & E = z \cdot \frac{σ}{\sqrt{n}} \\ σ = 1.5 & E = 1.645 \cdot (\frac{1.5}{\sqrt{200}}) \\ n = 200 & E = 0.1745 \end{matrix}$

Sample: For a 90% confidence level, the maximum error of the estimate is 0.17 days.

CAG found that the sample mean delivery time is 3.25 days. Construct the confidence interval (CI) for 90% confidence level.

$\begin{array}{rcl} \bar{x} - E & \leq & μ \leq \bar{x} + E \\ 3.25 - 0.1745 & \leq & μ \leq 3.25 + 0.1745 \\ 3.0755 & \leq & μ \leq 3.4245 \end{array}$

Note

It may be helpful to convert the decimal values to hours rather than a fractional part of 24 hours.

Q: How does the CI relate to the problem?

A: The CAG can be 90% confident that the population mean for delivery time is between 3.0755 and 3.4245 days.

Based on the CI, is Danell’s Delivery Service claim of 3.0 days plausible? Explain what this means for CAG.

Sample: The claim falls outside the range of values for the population mean, making it not plausible. CAG can be 90% confident that actual delivery times are slower than 3.0 days.

Danell’s Delivery Service noted that, if a 99% confidence level had been used, 3.0 days would have been within the CI. CAG said 90% is more helpful for customers. Explain why this was CAG’s response.

Sample: CAG used a 90% confidence level because it is more precise in delivery times. Customers want shorter delivery times for orders.

Note

Remember, a 90% confidence level is closer to the target (population mean). It is better to have a smaller window of time to wait for a delivery than a larger one.

Rather than using average delivery time, Danell’s Delivery Service decided to switch to customer satisfaction ratings. In a random survey of 750 customers, 87% reported being very satisfied. Determine the interval when the margin of error is $\pm 3.65 %$ .

$\begin{array}{l} 87 - 3.65 = 83.35 % \\ 87 + 3.65 = 90.65 % \end{array}$

Sample: It is likely that between 83.35% and 90.65% of customers will be satisfied with Danell’s Delivery Service.

Say What You Know

In your own words, talk about what you have learned using the objectives for this lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

Infer a population mean from confidence intervals.
Calculate the margin of error of a sample.
Evaluate reports.