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Chapter 9: Problem 65

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 \(0.8 \mathrm{lb}\).

Short Answer

Expert verified
The required sample size is around 246 as computed from the formula, rounded off to the nearest whole number.

Step by step solution

01

Identify the known variables

In the problem you can identify the following variables: \n- The standard deviation, \(\sigma\), which is 0.8 pounds.\n- The desired margin of error, E, which is 0.1 pounds.\n- The confidence level, which is 95%.
02

Determine the z-score corresponding to the given confidence level

For a confidence level of 95%, the value of \(Z_{\alpha/2}\) is approximately 1.96. This value can be found in standard statistical tables or calculated using a statistical software.
03

Apply the formula to find the needed sample size

Substitute the given values into the formula \(n = (Z_{\alpha/2}*\sigma/E)^2\). Here, \(Z_{\alpha/2}\) is 1.96, \(\sigma\) is 0.8, and E is 0.1. The calculation would look like this: \(n = (1.96*0.8/0.1)^2\). Calculate that to get the sample size required.
04

Round off the sample size

In practice, the sample size should be a whole number since you cannot collect a fraction of a sample. So, round off the computed sample size from the previous step to the nearest whole number, if necessary.

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Key Concepts

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

Margin of Error
The margin of error is a statistic expressing the amount of random sampling error in a survey's results. It represents the extent to which the estimated value may differ from the true population value due to randomness in sample selection. In the context of the textbook binding strength estimation, a margin of error of 0.1 lb implies that the true average force required to break the binding of the books is likely to fall within 0.1 lb of the estimated average force, either above or below it.

Understanding the margin of error is crucial for accurately interpreting study results. In practice, a smaller margin of error requires a larger sample size to increase the precision of the estimated value, as it suggests a tighter range around the estimated parameter. Conversely, a larger margin of error indicates a wider range and generally necessitates a smaller sample size.
Confidence Level
The confidence level is a measure of certainty or the probability with which we can expect a particular interval to capture the population parameter, assuming the study is repeated multiple times. A 95% confidence level, as mentioned in the textbook exercise, means that if we were to take 100 different samples and compute a confidence interval for each sample, we would expect 95 of those intervals to contain the true population mean.

When determining a sample size, the desired confidence level directly affects the width of the confidence interval; a higher confidence level results in a wider interval. This is because a higher confidence level requires more certainty that the population parameter is within the interval, necessitating a larger sample size. It is directly related to the critical value or z-score used in calculating the sample size, which determines how far the confidence interval extends from the sample mean.
Standard Deviation
Standard deviation, symbolized by the Greek letter sigma (\r( \( \sigma \))) in statistics, describes the amount of variation or dispersion in a set of values. A low standard deviation indicates the data points tend to be close to the mean of the set, while a high standard deviation indicates the data points are spread out over a wider range of values.

For the exercise concerning the strength of textbook bindings, the standard deviation is known and is 0.8 lb. This value is used to infer how varied the binding strength measurements are likely to be around the average force required to break the binding. In sample size determination, the standard deviation is a critical component as it aids in gauging the variability of the data. A larger standard deviation usually requires a larger sample size to achieve the same margin of error for a given confidence level, reflecting the increased variability and the need for more data to accurately estimate the population parameter.

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Most popular questions from this chapter

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