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Chapter 9: Problem 23

The trade magazine QSR routinely checks the drive-through service times of fast-food restaurants. A \(90 \%\) confidence interval that results from examining 607 customers in Taco Bell's drive-through has a lower bound of 161.5 seconds and an upper bound of 164.7 seconds. What does this mean?

Short Answer

Expert verified
We are 90% confident that the true average drive-through service time at Taco Bell is between 161.5 and 164.7 seconds.

Step by step solution

01

Identify the Confidence Interval

The confidence interval given is from 161.5 seconds to 164.7 seconds, which includes all the likely values of the population parameter (mean drive-through time) based on the sample.
02

Understand Confidence Level

The confidence level provided is 90%. This indicates that we are 90% confident that the true mean drive-through time for all customers at Taco Bell lies within the confidence interval.
03

Interpret Lower and Upper Bounds

The lower bound of the interval is 161.5 seconds and the upper bound is 164.7 seconds. Hence, the interval suggests that the actual average service time is very likely between these two values.

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Key Concepts

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

confidence level
When we talk about the confidence level, we refer to the degree of certainty we have that a given parameter falls within the specified confidence interval.
The confidence level is often expressed as a percentage. In this exercise, it is stated as 90%.
This means that if we took 100 different samples and calculated a confidence interval for each sample, then approximately 90 of those intervals would contain the true mean.
So, a 90% confidence level indicates high reliability, but not absolute certainty.
This helps us understand that our interval is highly likely to include the true mean drive-through time.
lower bound
The lower bound is the smallest value in our confidence interval.
In this exercise, the lower bound is 161.5 seconds. This means that at the lowest end, we are confident that the mean drive-through time won't fall below this value by very much.
It represents the minimum average time you can expect with 90% confidence.
Lower bounds are essential as they provide a range's starting limit, helping us understand the worst-case scenario in terms of time efficiency.
upper bound
The upper bound is the highest value in our confidence interval.
Here, it鈥檚 at 164.7 seconds.
It indicates the maximum average time the drive-through might take according to our sample, likewise with 90% confidence.
Upper bounds serve to cap the range, highlighting the most extended drive-through time we can reasonably expect.
Knowing the upper bound helps in assessing the best-case scenario policy changes or improvements.
mean drive-through time
The mean drive-through time refers to the average time it takes for a customer to get through the drive-through.
In statistical terms, it is represented by the population parameter often denoted as \(\bar{X}\).
This exercise's confidence interval hints that the true mean lies between 161.5 and 164.7 seconds.
It鈥檚 essential to calculate this mean because it helps businesses understand their average service efficiency and identify any potential delays or improvements needed.
sample size
Sample size is the number of observations used to compute the confidence interval and other statistics.
In our exercise, the sample size is 607.
Larger sample sizes generally lead to more accurate estimates of the population parameters because they tend to reduce the margin of error.
With a sample size of 607, the confidence interval becomes more reliable, making the conclusions about the mean drive-through time more robust.
The larger the sample, the smaller the uncertainty, and that鈥檚 why having a big enough sample size is crucial in statistics.

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Most popular questions from this chapter

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