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Chapter 9: Problem 14

Discuss how each of the following factors affects the width of the confidence interval for \(\pi\) : a. The confidence level b. The sample size c. The value of \(p\)

Short Answer

Expert verified
The confidence level and the width of the confidence interval are directly proportional 鈥� as one increases, so does the other. Sample size is inversely proportional to the width of the confidence interval 鈥� as sample size increases, the interval decreases. The value of \(p\) affects the interval width, but not linearly 鈥� the width is largest when \(p = 0.5\) and gets narrower as \(p\) approaches 0 or 1.

Step by step solution

01

Effect of Confidence Level

The confidence level is correlated with the width of the confidence interval. As the confidence level increases, the z-score also increases. This is because a higher confidence level means the interval needs to capture the true population parameter, \(\pi\), more frequently, so the interval must be wider. Thus, the confidence level is directly proportional to the width of the confidence interval 鈥� as one increases, so does the other.
02

Effect of Sample Size

The sample size has an inverse relationship with the width of the confidence interval. As the sample size increases, the standard error decreases as it is calculated as \( \sqrt{pq/n} \), where \( n \) is the sample size, and \( p \) and \( q \) are the estimated proportions of the population. Because the standard error is in the denominator, a larger sample size means a smaller standard error, and thus a narrower confidence interval.
03

Effect of Value of \(p\)

The value of \(p\) affects the width of the confidence interval but the relationship is not linear. The standard error is at its maximum when \(p = 0.5\). As \(p\) moves away from 0.5 in either direction (towards 0 or 1), the standard error decreases and so does the width of the confidence interval.

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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 an indication of how sure we are that the true parameter lies within our confidence interval. If you choose a higher confidence level, say 95% instead of 90%, you are expressing a desire to be more certain. To achieve this, the interval must widen to cover more possible values of the true population proportion, \( \pi \).
  • A higher confidence level means a wider confidence interval.
  • This is because the z-score increases as the confidence level rises.
  • A wider interval captures the true parameter more reliably.
In summary, increasing your confidence level will widen your confidence interval because you're asking for more assurance that it includes the true value.
Sample Size
The sample size plays a crucial role in determining the width of a confidence interval. It is inversely related, meaning that as the sample size grows, the confidence interval becomes narrower. This is due to the standard error component in the confidence interval formula, \( \text{SE} = \sqrt{\frac{pq}{n}} \).
  • Larger sample sizes mean smaller standard errors.
  • Smaller standard errors lead to narrower confidence intervals.
  • This provides more precise estimates of the population proportion.
So, remember: increasing the sample size shrinks the confidence interval, giving you a more accurate estimate of \( \pi \). This is why surveys or studies aim for larger participant numbers when possible.
Proportion Value
The proportion value, represented as \( p \), influences the width of the confidence interval, but the effect is non-linear. The formula for standard error includes \( p \) and \( q \) (where \( q = 1 - p \), the complement of \( p \)). The maximum standard error occurs when \( p = 0.5 \).
  • When \( p \) is close to 0.5, the standard error is larger, resulting in a wider confidence interval.
  • Moving \( p \) towards 0 or 1 decreases the standard error, narrowing the interval.
  • This happens because extremes of 0 or 1 offer more certainty about the population proportion.
Overall, the value of \( p \) affects the interval width, but keep in mind that it's widest when \( p \) is 0.5. It's fascinating how the middle value of the range provides the least precision due to greater variability.

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Most popular questions from this chapter

Retailers report that the use of cents-off coupons is increasing. The Scripps Howard News Service (July 9 , 1991) reported the proportion of all households that use coupons as .77. Suppose that this estimate was based on a random sample of 800 households (i.e., \(n=800\) and \(p=.77\) ). Construct a \(95 \%\) confidence interval for \(\pi\), the true proportion of all households that use coupons.

In a study of 1710 schoolchildren in Australia (Herald Sun, October 27,1994 ), 1060 children indicated that they normally watch TV before school in the morning. (Interestingly, only \(35 \%\) of the parents said their children watched TV before school!) Construct a \(95 \%\) confidence interval for the true proportion of Australian children who say they watch TV before school. What assumption about the sample must be true for the method used to construct the interval to be valid?

The article "Consumers Show Increased Liking for Diesel Autos" (USA Today, January 29,2003 ) reported that \(27 \%\) of U.S. consumers would opt for a diesel car if it ran as cleanly and performed as well as a car with a gas engine. Suppose that you suspect that the proportion might be different in your area and that you want to conduct a survey to estimate this proportion for the adult residents of your city. What is the required sample size if you want to estimate this proportion to within \(.05\) with \(95 \%\) confidence? Compute the required sample size first using 27 as a preliminary estimate of \(\pi\) and then using the conservative value of \(.5 .\) How do the two sample sizes compare? What sample size would you recommend for this study?

Given a variable that has a \(t\) distribution with the specified degrees of freedom, what percentage of the time will its value fall in the indicated region? a. \(10 \mathrm{df}\), between \(-1.81\) and \(1.81\) b. \(10 \mathrm{df}\), between \(-2.23\) and \(2.23\) c. \(24 \mathrm{df}\), between \(-2.06\) and \(2.06\) d. \(24 \mathrm{df}\), between \(-2.80\) and \(2.80\) e. 24 df, outside the interval from \(-2.80\) to \(2.80\) f. \(24 \mathrm{df}\), to the right of \(2.80\) g. \(10 \mathrm{df}\), to the left of \(-1.81\)

A study of the ability of individuals to walk in a straight line ("Can We Really Walk Straight?" American Journal of Physical Anthropology [1992]: \(19-27\) ) reported the following data on cadence (strides per second) for a sample of \(n=20\) randomly selected healthy men: \(\begin{array}{llllllllll}0.95 & 0.85 & 0.92 & 0.95 & 0.93 & 0.86 & 1.00 & 0.92 & 0.85 & 0.81\end{array}\) \(\begin{array}{llllllllll}0.78 & 0.93 & 0.93 & 1.05 & 0.93 & 1.06 & 1.06 & 0.96 & 0.81 & 0.96\end{array}\) Construct and interpret a \(99 \%\) confidence interval for the population mean cadence.
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