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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

Blood Alcohol Concentration A random sample of 51 fatal crashes in 2013 in which the driver had a positive blood alcohol concentration (BAC) from the National Highway Traffic Safety Administration results in a mean BAC of 0.167 gram per deciliter \((\mathrm{g} / \mathrm{dL})\) with a standard deviation of \(0.010 \mathrm{~g} / \mathrm{dL}\) (a) A histogram of blood alcohol concentrations in fatal accidents shows that BACs are highly skewed right. Explain why a large sample size is needed to construct a confidence interval for the mean BAC of fatal crashes with a positive \(\mathrm{BAC}\) (b) In \(2013,\) there were approximately 25,000 fatal crashes in which the driver had a positive BAC. Explain why this, along with the fact that the data were obtained using a simple random sample, satisfies the requirements for constructing a confidence interval. (c) Determine and interpret a \(90 \%\) confidence interval for the mean BAC in fatal crashes in which the driver had a positive BAC. (d) All 50 states and the District of Columbia use a BAC of \(0.08 \mathrm{~g} / \mathrm{dL}\) as the legal intoxication level. Is it possible that the mean BAC of all drivers involved in fatal accidents who are found to have positive BAC values is less than the legal intoxication level? Explain.

Determine the point estimate of the population proportion, the margin of error for each confidence interval, and the number of individuals in the sample with the specified characteristic, \(x,\) for the sample size provided. Lower bound: \(0.853,\) upper bound: \(0.871, n=10,732\)

True or False: To construct a confidence interval about the mean, the population from which the sample is drawn must be approximately normal.

Explain why quadrupling the sample size causes the margin of error to be cut in half.

A sociologist wishes to conduct a poll to estimate the percentage of Americans who favor affirmative action programs for women and minorities for admission to colleges and universities. What sample size should be obtained if she wishes the estimate to be within 4 percentage points with \(90 \%\) confidence if (a) she uses a 2003 estimate of \(55 \%\) obtained from a Gallup Youth Survey? (b) she does not use any prior estimates? (c) Why are the results from parts (a) and (b) so close?
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