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Chapter 8: Problem 78

a. How large a sample should be selected so that the margin of error of estimate for a \(99 \%\) confidence interval for \(p\) is \(.035\) when the value of the sample proportion obtained from a preliminary sample is \(.29 ?\) b. Find the most conservative sample size that will produce the margin of error for a \(99 \%\) confidence interval for \(p\) equal to \(.035\).

Short Answer

Expert verified
The required sample size when the sample proportion is given (.29) is roughly 756. The conservative sample size with 99% confidence level and a margin of error of 0.035 is approximately 1082.

Step by step solution

01

Calculate the Sample Size for Given Sample Proportion

Insert the provided values into the formula for the margin of error, and rearrange the formula to solve for the sample size, \(n\). The formula becomes \(n = (z^2 \cdot p \cdot (1-p)) / E^2\). Plugging in \(z = 2.575\), \(p = .29\) and \(E = .035\), \(n = ((2.575^2) \cdot .29 \cdot (1-.29)) / .035^2 = 755.81 \). Because you can't have a fraction of a person, round up to \(n = 756\).
02

Find the Conservative Sample Size for Given Margine of Error

When no preliminary estimate is available, use 0.5 for \(p\) (as it will maximize the product \(p(1-p)\)). So, with \(p = .5\), \(z = 2.575\), and \(E = .035\), plug into the formula as in Step 1. The formula becomes \(n = ((2.575^2) \cdot .5 \cdot (1-.5))/ .035^2 = 1081.35 \approx 1082 \) (again, rounding up to the next whole number).

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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 that is used to estimate an unknown population parameter. It provides an interval within which we expect the true parameter to lie, with a certain level of confidence. For example, when we say a confidence interval of 99%, it implies that if we were to take 100 different samples and compute a confidence interval for each sample, we expect roughly 99 of them to contain the true population parameter.
To form a confidence interval, you need:
  • A sample statistic, like the sample mean or proportion
  • The margin of error
  • A confidence level (e.g., 99%)
Understanding confidence intervals helps in making informed decisions based on statistical data, giving you a buffer of certainty during estimation.
Margin of Error
The margin of error indicates the extent of uncertainty associated with a sample statistic as an estimate of a population parameter. In simpler terms, it expresses how much the statistics from your sample might differ from the actual population value.
The margin of error depends on:
  • Sample size: Larger samples tend to have a smaller margin of error.
  • Confidence level: Higher confidence levels increase the margin of error.
  • Sample variability: More variability in your data means a larger margin of error.
In the original problem, you are specifically calculating the sample size required to achieve a margin of error of 0.035 in a 99% confidence interval, showing the delicate balance between precision and confidence.
Sample Proportion
Sample proportion refers to the percentage or fraction of the sample that has a particular attribute of interest. It is calculated by dividing the number of favorable outcomes by the total number of samples.
For example, if you survey 100 people and 29 of them prefer a specific product, the sample proportion would be 0.29 (or 29%).
The sample proportion is used in statistical estimation to make generalizations about a population. It plays a crucial role in determining:
  • The confidence interval and margin of error
  • The necessary sample size for statistical reliability
In the given exercise, the sample proportion was 0.29, which was used to calculate the sample size.
Statistical Estimation
Statistical estimation involves making inferences about a population based on data obtained from a sample. This process is fundamental for predicting and understanding population parameters without the need to survey everyone.
There are two main types of statistical estimation:
  • Point estimation: A single number estimate of a population parameter.
  • Interval estimation: A range of values used to estimate the parameter.
In the exercise, statistical estimation is applied through determining both a specific sample size and a more conservative estimate to achieve desired precision in the form of a margin of error. This ensures that decisions are backed by mathematically sound reasoning and provide confidence in the results.

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Most popular questions from this chapter

It is said that happy and healthy workers are efficient and productive. A company that manufactures exercising machines wanted to know the percentage of large companies that provide on-site health club facilities. A sample of 240 such companies showed that 96 of them provide such facilities on site. a. What is the point estimate of the percentage of all such companies that provide such facilities on site? b. Construct a \(97 \%\) confidence interval for the percentage of all such companies that provide such facilities on site. What is the margin of error for this estimate?

A computer company that recently developed a new software product wanted to estimate the mean time taken to learn how to use this software by people who are somewhat familiar with computers. A random sample of 12 such persons was selected. The following data give the times taken (in hours) by these persons to learn how to use this software. \(\begin{array}{llllll}1.75 & 2.25 & 2.40 & 1.90 & 1.50 & 2.75 \\ 2.15 & 2.25 & 1.80 & 2.20 & 3.25 & 2.60\end{array}\) Construct a \(95 \%\) confidence interval for the population mean. Assume that the times taken by all persons who are somewhat familiar with computers to learn how to use this software are approximately normally distributed.

For each of the following, find the area in the appropriate tail of the \(t\) distribution. a. \(t=2.467\) and \(d f=28\) b. \(t=-1.672\) and \(d f=58\) c. \(t=-2.670\) and \(n=55\) d. \(t=2.819\) and \(n=23\)

A survey of 500 randomly selected adult men showed that the mean time they spend per week watching sports on television is \(9.75\) hours with a standard deviation of \(2.2\) hours. Construct a \(90 \%\) confidence interval for the population mean, \(\mu .\)

n April 2012, N3L Optics conducted a telephone poll of 1080 adult Americans aged 18 years and older. One of the questions asked respondents to identify which outdoor activities and sports they favor for fitness. Respondents could choose more than one activity/sport. Of the respondents, \(76 \%\) said walking, \(35 \%\) mentioned hiking, and \(27 \%\) said team sports (http://n3loptics.com/news_items/57). Using these results, find a \(98 \%\) confidence interval for the population percentage that corresponds to each response. Write a one-page report to present your results to a group of college students who have not taken statistics. Your report should answer questions such as the following: (1) What is a confidence interval? (2) Why is a range of values (interval) more informative than a single percentage (point estimate)? (3) What does \(98 \%\) confidence mean in this context? (4) What assumptions, if any, are you making when you construct each confidence interval?
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