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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
The width of a confidence interval decreases with a decrease in the confidence level because these two parameters are inversely related. For instance, if the 95% confidence interval for a population mean is (40, 60), decreasing the confidence level to 90% might result in a narrower interval like (45, 55), depending on the data.

Step by step solution

01

Understanding Confidence Intervals

A confidence interval is a range of values, derived from a data sample, that is likely to contain the value of an unknown parameter. It is usually reported in conjunction with a confidence level that, loosely speaking, quantifies the level of confidence that the parameter lies within the interval. Examples of confidence level are 90%, 95%, and 99%.
02

Relationship between Confidence Level and Width

The confidence level and width of the confidence interval are inversely related. As the confidence level decreases, our certainty about the estimate also decreases. Consequently, to maintain this lower level of confidence, the range (or width) of the interval can be narrowed. Conversely, to maintain a high level of confidence, we would require a wider CONFIDENCE interval.
03

Example

For example, consider a population mean, and suppose the 95% confidence interval for the mean is (40, 60). This means that we are 95% confident that the true mean lies within this interval. Now, if we lower the confidence level to 90%, the level of certainty decreases. Therefore, to maintain this reduced level of confidence, we can narrow the interval, say to (45, 55).

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Key Concepts

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

Confidence Level
The confidence level is the probability that a statistical estimate like a confidence interval contains the true population parameter. When we talk about a 95% confidence level, we mean that if we were to take 100 different samples and compute a confidence interval from each sample, we would expect about 95 of those intervals to contain the true parameter, such as the population mean.
  • Common confidence levels are 90%, 95%, and 99%.
  • A higher confidence level means more certainty that the interval contains the true parameter.
  • However, as the confidence level increases, the width of the confidence interval also increases.
Population Mean
The population mean is the average of all values in a population and is a measure of the central tendency. In many studies, the population mean is unknown, and researchers use a sample to estimate it. This is where confidence intervals come into play. They provide a range in which the true population mean is likely to be found, given the data from your sample.
  • Population mean is often denoted by the Greek letter 渭.
  • Sample means are used to make inferences about the population mean.
  • Using a sample is often necessary because measuring the entire population is impractical.
Confidence Interval Width
The width of a confidence interval is influenced by several factors, including the confidence level and the variability of the data. As seen in the original exercise, there's an inverse relationship between confidence level and interval width:
  • Lower confidence levels lead to narrower confidence intervals.
  • Higher confidence levels require wider intervals to maintain certainty.
  • Interval width is also affected by the size of the sample and variability within it.
For example, with a 95% confidence interval of (40, 60), we're 95% sure the true mean lies in that range. Lowering to a 90% confidence level narrows it to (45, 55), as less certainty allows a tighter range.
Statistical Estimate
A statistical estimate is an approximation of a population parameter based on sample data. There are two types of estimates: point estimates and interval estimates. A point estimate is a single value used to approximate a parameter, such as the sample mean for the population mean. Meanwhile, an interval estimate provides a range of valuesx鈥攖his is what a confidence interval represents.
  • Estimates are essential for making informed conclusions about populations.
  • The accuracy of a statistical estimate is often reflected in the confidence interval.
  • Estimates become more reliable as sample size increases.
In sum, while a point estimate gives a specific number, interval estimates like confidence intervals provide more information by allowing for variability and uncertainty that inherently exists in sample data.

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Most popular questions from this chapter

A bank manager wants to know the mean amount of mortgage paid per month by homeowners in an area. A random sample of 120 homeowners selected from this area showed that they pay an average of \(\$ 1575\) per month for their mortgages. The population standard deviation of such mortgages is \(\$ 215\). a. Find a \(97 \%\) confidence interval for the mean amount of mortgage paid per month by all homeowners in this area. b. Suppose the confidence interval obtained in part a is too wide. How can the width of this interval be reduced? Discuss all possible alternatives. Which alternative is the best?

For a data set obtained from a sample, \(n=20\) and \(\bar{x}=24.5\). It is known that \(\sigma=3.1\). The population is normally distributed. a. What is the point estimate of \(\mu\) ? b. Make a \(99 \%\) confidence interval for \(\mu\). c. What is the margin of error of estimate for part b?

You are interested in estimating the mean commuting time from home to school for all commuter students at your school. Briefly explain the procedure you will follow to conduct this study. Collect the required data from a sample of 30 or more such students and then estimate the population mean at a \(99 \%\) confidence level. Assume that the population standard deviation for such times is \(5.5\) minutes.

When one is attempting to determine the required sample size for estimating a population mean, and the information on the population standard deviation is not available, it may be feasible to take a small preliminary sample and use the sample standard deviation to estimate the required sample size, \(n\). Suppose that we want to estimate \(\mu\), the mean commuting distance for students at a community college, to a margin of error within 1 mile with a confidence level of \(95 \%\). A random sample of 20 students yields a standard deviation of \(4.1\) miles. Use this value of the sample standard deviation, \(s\), to estimate the required sample size, \(n\). Assume that the corresponding population has a normal distribution.

At the end of Section \(8.2\), we noted that we always round up when calculating the minimum sample size for a confidence interval for \(\mu\) with a specified margin of error and confidence level. Using the formula for the margin of error, explain why we must always round up in this situation.
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