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Chapter 7: Problem 21

In order to halve the width of a \(95 \%\) confidence interval for a mean, by what factor should the sample size be increased? Ignore the finite population correction.

Short Answer

Expert verified
Increase the sample size by a factor of 4.

Step by step solution

01

Understanding Confidence Interval Width

The width of a confidence interval is determined by the formula: \[2 imes Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}\]where \(Z_{\alpha/2}\) is the z-value, \(\sigma\) is the standard deviation, and \(n\) is the sample size. Our goal is to halve the width of this confidence interval.
02

Setting Up the Equation

Let's denote the original sample size as \(n\), and the original interval width as \(W\). To halve this width, we need the new width to be \(\frac{W}{2}\). The original width formula is: \[W = 2 \times Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}\]The new width should be: \[\frac{W}{2} = 2 \times Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n'}}\]where \(n'\) is the new sample size.
03

Solving for New Sample Size

Set the equations equal since we want the interval to be half, \[2 \times Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n'}} = \frac{1}{2}\left(2 \times Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}\right)\]By simplifying, we get:\[\frac{\sigma}{\sqrt{n'}} = \frac{\sigma}{2\sqrt{n}}\]which simplifies to \[\sqrt{n'} = 2\sqrt{n}\]
04

Solving for Factor

Square both sides to solve for the new sample size:\[n' = 4n\]Thus, the sample size must be increased by a factor of 4 to halve the confidence interval width.

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Key Concepts

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

Confidence Interval
A confidence interval is a range of values used to estimate a population parameter, such as the mean. It is associated with a specific confidence level, often expressed as a percentage, like 95%. This percentage indicates how certain we are that the true parameter lies within the interval. The width of the confidence interval is crucial because it tells us about the precision of our estimate. A narrower confidence interval provides a more precise estimate of the parameter. However, achieving this precision often requires having a larger sample size. Thus, understanding and manipulating confidence intervals involves balancing between desired precision and the practicalities of data collection.
Statistical Mean
The statistical mean, often referred to simply as the mean, is a measure of central tendency. It sums up all the data points in a dataset and divides by the number of points. This concept is important in understanding sample means and how they relate to population means. When we calculate the mean from a sample, it serves as an estimate of the population mean. The reliability of this estimate improves as the sample size increases, which also leads to a more reliable confidence interval. The mean is therefore a fundamental concept when dealing with statistics, forming the foundation for other statistical metrics.
Sample Size Calculation
Determining the right sample size is vital for drawing reliable conclusions. Sample size calculation involves understanding how much data is needed to accurately represent the population. Increasing the sample size reduces the margin of error, resulting in a narrower confidence interval and more precise estimates. As seen in typical problems, sometimes you may need to increase the sample size by a considerable factor to achieve the desired confidence interval width, like multiplying by 4 to reduce the width by half. In short, careful sample size planning is crucial for effective statistical analysis and for obtaining meaningful insights.
Z-Score
A Z-score is a measure of how many standard deviations an element is from the mean. In the context of confidence intervals, the Z-score associated with the desired confidence level is crucial. For example, a 95% confidence level has a Z-score of approximately 1.96. The Z-score helps to determine the width of the confidence interval by scaling the standard error of the mean. Thus, it plays a pivotal role in shaping the confidence interval. Understanding how to use Z-scores in statistical analysis is essential when wanting to make confident assertions about population parameters.

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Most popular questions from this chapter

Which of the following is a random variable? a. The population mean b. The population size, \(N\) c. The sample size, \(n\) d. The sample mean e. The variance of the sample mean f. The largest value in the sample g. The population variance h. The estimated variance of the sample mean

How would you respond to a friend who asks you, "How can we say that the sample mean is a random variable when it is just a number, like the population mean? For example, in Example A of Section 7.3.2, a simple random sample of size 50 produced \(\bar{x}=938.5 ;\) how can the number 938.5 be a random variable?鈥�

This problem presents an algorithm for drawing a simple random sample from a population in a sequential manner. The members of the population are considered for inclusion in the sample one at a time in some prespecified order (for example, the order in which they are listed). The \(i\) th member of the population is included in the sample with probability \(\frac{n-n_{i}}{N-i+1}\) "where \(n_{i}\) is the number of population members already in the sample before the ith member is examined. Show that the sample selected in this way is in fact a simple random sample; that is, show that every possible sample occurs with probability $$\frac{1}{\left(\begin{array}{l}N \\\n\end{array}\right)}$$

In the population of hospitals, the correlation of the number of beds and the number of discharges is \(\rho=.91 \text { (Example D of Section } 7.4) .\) To see how \(\operatorname{Var}\left(\bar{Y}_{R}\right)\) would be different if the correlation were different, plot \(\operatorname{Var}\left(\bar{Y}_{R}\right)\) for \(n=64\) as a function of \(\rho\) for \(-1<\rho<1\)

(Computer Exercise) Construct a population consisting of the integers from 1 to 100\. Simulate the sampling distribution of the sample mean of a sample of size 12 by drawing 100 samples of size 12 and making a histogram of the results.
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