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Chapter 9: Problem 11

would result in a wider large-sample cont?dence interval for \(\pi\) : a. \(90 \%\) confidence level or \(95 \%\) confidence level b. \(n=100\) or \(n=400\)

Short Answer

Expert verified
The wider large-sample confidence interval for \(\pi\) will be resulted from \(95\%\) confidence level (compared to \(90\%\) confidence level) and \(n=100\) (compared to \(n=400\)).

Step by step solution

01

Understand Confidence Level Effect

Confidence intervals denote the range in which the true population parameter lies with a certain confidence level. As the confidence level increases, the range of confidence intervals also expands. Hence, a \(95\%\) confidence level will provide a wider confidence interval as compared to a \(90\%\) confidence level.
02

Understand Sample Size Effect

Sample size plays an important role in determining the width of confidence intervals. The larger the sample size (n), the more narrowed the confidence interval. That's due to the smaller standard error with a larger sample size. Hence, \(n=100\) will yield a wider confidence interval compared to \(n=400\).
03

Combine The Concepts

By combining both concepts you can conclude that a \(95\%\) confidence interval will always be larger than a \(90\%\) one and a confidence interval based on a smaller sample (like \(n=100\)) will be larger than one based on a large sample (like \(n=400\)).

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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're referring to how certain we are that the confidence interval contains the true population parameter. Imagine confidence levels like trust measures. A higher confidence level means you're more certain you're capturing the right data, but there's a twist. To increase your confidence, you need a bigger net.
For instance, a 95% confidence level indicates you are 95% sure the true parameter is within the interval. But since you're more confident, this interval stretches out a bit more compared to a 90% confidence level. In simple terms:
  • A 90% confidence level is narrower, meaning less confident but tighter.
  • A 95% confidence level is wider, offering more certainty but a broader scope.
Understanding this helps you decide how accurate or wide-ranging you want your data interval to be.
Sample Size
Sample size is like the number of snapshots you take from a population. Think of it as the strength of your data collection method. A larger sample size gives you a better and more reliable look at the entire data landscape.
The relationship between sample size and confidence interval width is straightforward: more data means less guessing, leading to a narrower interval.
  • Large sample size ( =400) results in a tight, more accurate interval.
  • Small sample size ( =100) yields a wider, less precise interval.
The reason behind this is the concept of variability. A larger sample size reduces variability because you have more information, getting you closer to the true value. Therefore, when planning a study, it鈥檚 crucial to balance the cost of larger samples with the improved precision they bring.
Standard Error
Standard error is the hero behind the scenes in statistics, helping gauge the accuracy of sample means. It tells us how far off our sample mean might be from the true population mean.
Think of it as a measure of statistical reliability. The standard error decreases with increasing samples, and here's why it matters:
  • Smaller standard error means your sample mean is likely a good reflection of the population mean.
  • Larger standard error suggests more inherent variability, which can widen the confidence interval.
How do we calculate it? It's dependent on two factors:
  • The standard deviation of the sample, representing inherent data spread.
  • The square root of the sample size, which stabilizes as the sample size grows.
Ultimately, a smaller standard error, achieved through a larger sample, leads to a more precise confidence interval, crucial for making confident inferences about a population.

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Most popular questions from this chapter

A manufacturer of college textbooks is interested in estimating the strength of the bindings produced by a particular binding machine. Strength can be measured by recording the force required to pull the pages from the binding. If this force is measured in pounds, how many books should be tested to estimate with \(95 \%\) confidence to within \(0.1 \mathrm{lb}\), the average force required to break the binding? Assume that \(\sigma\) is known to be \(0.8 \mathrm{lb}\).

Conducts an annual survey 6 of all households expe the past year. This esti102 randomly selected adults. The report states, "One can say with \(95 \%\) confidence that the margin of sampling error is \(\pm 3\) percentage points." Explain how this statement can be justified.

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?

Five students visiting the student health center for a free dental examination during National Dental Hygiene Month were asked how many months had passed since their last visit to a dentist. Their responses were as follows: \(\begin{array}{lllll}6 & 17 & 11 & 22 & 29\end{array}\) Assuming that these five students can be considered a random sample of all students participating in the free checkup program, construct a \(95 \%\) confidence interval for the mean number of months elapsed since the last visit to a dentist for the population of students participating in the program.

The article "Sensory and Mechanical Assessment of the Quality of Frankfurters" (Journal of Texture Studies [1990]: \(395-409\) ) reported the following salt content (percentage by weight) for 10 frankfurters: \(\begin{array}{llllllllll}2.26 & 2.11 & 1.64 & 1.17 & 1.64 & 2.36 & 1.70 & 2.10 & 2.19 & 2.40\end{array}\) a. Use the given data to produce a point estimate of \(\mu\), the true mean salt content for frankfurters. b. Use the given data to produce a point estimate of \(\sigma^{2}\), the variance of salt content for frankfurters. c. Use the given data to produce an estimate of \(\sigma\), the standard deviation of salt content. Is the statistic you used to produce your estimate unbiased?
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