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Chapter 8: Problem 6

Briefly explain how the width of a confidence interval decreases with a decrease in the confidence level. Give an example.

Short Answer

Expert verified
In summary, the width of a Confidence Interval decreases with a decrease in the Confidence Level because a lower Confidence Level means we are considering a narrower range of values for our unknown population parameter.

Step by step solution

01

Definition of Confidence Interval and Confidence Level

Understand that a Confidence Interval (CI) is a range of values, derived from a data set, which is likely to contain the value of an unknown population parameter. The Confidence Level (CL) is the likelihood that the CI contains the unknown population parameter. So if we decrease the CL, this essentially decreases the range of values we are considering to include our unknown parameter, which in turn decreases the CI width.
02

Relation between Confidence Interval and Confidence Level

It's important to emphasize that as CL decreases, so does the width of the CI. This is because lower CL indicates less certainty about the unknown parameter's location, thereby a tighter range of values (smaller CI) is to be expected.
03

Example

For example, let's consider we are to estimate the average height of students in a school. Let's say the 95% CI for average height is (160cm, 180cm). This CI tells us that we are 95% confident that the true population parameter (average height) lies within this range. If we reduce the CL to 90%, we are less confident about where average height lies, and hence our CI will decrease, say for example to (165cm, 175cm).

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Most popular questions from this chapter

A businesswoman is considering whether to open a coffee shop in a local shopping center. Before making this decision, she wants to know how much money, on average, people spend per week at coffee shops in that area. She took a random sample of 26 customers from the area who visit coffee shops and asked them to record the amount of money (in dollars) they would spend during the next week at coffee shops. At the end of the week, she obtained the following data (in dollars) from these 26 customers: \(\begin{array}{rrrrrrrrr}16.96 & 38.83 & 15.28 & 14.84 & 5.99 & 64.50 & 12.15 & 14.68 & 33.37 \\ 37.10 & 18.15 & 67.89 & 12.17 & 40.13 & 5.51 & 8.80 & 34.53 & 35.54 \\ 8.51 & 37.18 & 41.52 & 13.83 & 12.96 & 22.78 & 5.29 & 9.09 & \end{array}\) Assume that the distribution of weekly expenditures at coffee shops by all customers who visit coffee shops in this area is approximately normal. a. What is the point estimate of the corresponding population mean? b. Make a \(95 \%\) confidence interval for the average amount of money spent per week at coffee shops by all customers who visit coffee shops in this area.

Determine the sample size for the estimate of \(\mu\) for the following. a. \(E=.17, \quad \sigma=.90\), confidence level \(=99 \%\) b. \(E=1.45, \quad \sigma=5.82, \quad\) confidence level \(=95 \%\) c. \(E=5.65, \quad \sigma=18.20, \quad\) confidence level \(=90 \%\)

A consumer agency wants to estimate the proportion of all drivers who wear seat belts while driving. Assume that a preliminary study has shown that \(76 \%\) of drivers wear seat belts while driving. How large should the sample size be so that the \(99 \%\) confidence interval for the population proportion has a margin of error of \(.03\) ?

An economist wants to find a \(90 \%\) confidence interval for the mean sale price of houses in a state. How large a sample should she select so that the estimate is within \(\$ 3500\) of the population mean? Assume that the standard deviation for the sale prices of all houses in this state is \(\$ 31,500\).

a. Find the value of \(t\) for the \(t\) distribution with a sample size of 21 and area in the left tail equal to \(.10 .\) b. Find the value of \(t\) for the \(t\) distribution with a sample size of 14 and area in the right tail equal to \(.025 .\) c. Find the value of \(t\) for the \(t\) distribution with 45 degrees of freedom and \(.001\) area in the right tail. d. Find the value of \(t\) for the \(t\) distribution with 37 degrees of freedom and \(.005\) area in the left tail.
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