/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 67 A manufacturer of college textbo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 67

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 the mean force required to break the binding to within 0.1 pounds with \(95 \%\) conficence? Assume that \(\sigma\) is known to be 0.8 pound.

Short Answer

Expert verified
The manufacturer needs to test 246 books to estimate the mean force required to break the binding within 0.1 pounds with a 95% confidence level.

Step by step solution

01

Identify the given values and required value

From the problem, the known standard deviation (\(\sigma\)) is 0.8 pound. The margin of error (E) is 0.1 pound. The level of confidence is \(95\%\) which corresponds to a z-score of 1.96 (based on the Z-table or norm distribution table). The aim is to determine the sample size (n) to estimate the mean force.
02

Use the formula for sample size

The required sample size (n) in an estimate can be calculated using the formula: \(n = \( \left( Z * \frac{\sigma}{E} \right) ^2\), where Z is the z-score for the desired confidence level, \(\sigma\) is the known standard deviation, and E is the maximum allowable error.
03

Substitute the known values into the formula

Substitute the values into the formula to solve for n: \(n = \( \left( 1.96 * \frac{0.8}{0.1} \right) ^2\)
04

Solve for n

Computing the equation gives, \(n = 245.8624\)
05

Round up to the nearest whole number

Since we can't have a fractional number of books, we round up the result to nearest whole number, which give us \(n = 246.\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Confidence Interval
A confidence interval allows us to estimate the range within which the true population parameter lies, with a particular degree of certainty. In this problem, we are given a confidence level of 95%, which means we are 95% confident that the true mean force required to break the book binding lies within our calculated interval.

The confidence interval depends on several factors:
  • The sample size: Larger samples provide more information, yielding more precise estimates.
  • The standard deviation of the population: This tells us how much the data is spread out.
  • The chosen confidence level: Higher confidence levels widen the interval.
Understanding confidence intervals helps us understand the reliability of estimates, guiding decisions based on statistical data.
Standard Deviation
Standard deviation is a key concept in statistics that measures the amount of variation or dispersion in a set of values. In this problem, the standard deviation is known to be 0.8 pounds.

A smaller standard deviation indicates that the data points tend to be closer to the mean of the set, while a larger standard deviation indicates that the data points are spread out over a wider range of values. Standard deviation is critical in calculating the sample size, as it directly affects the width of the confidence interval:
  • If the standard deviation is high, it suggests that more data is required to achieve a precise estimate.
  • A smaller standard deviation would require fewer samples to understand the population parameter accurately.
By understanding standard deviation, we can better grasp the natural variability in measurements and make better predictions and decisions.
Margin of Error
The margin of error shows the range in which the true population parameter could lie, based on our sample's findings. In the textbook problem, we want the mean force estimation to be precise within 0.1 pounds.

The margin of error depends on several factors:
  • The chosen confidence level: Higher levels increase the margin of error, as we become more cautious in ensuring the true value lies within the range.
  • The standard deviation: More variability in the data naturally increases the margin of error.
  • The sample size: Larger samples can reduce the margin of error, making your estimates more precise.
Having a tight margin of error is crucial for reliability in data-driven decisions, allowing us to have greater confidence in the results.
Z-Score
A z-score represents how many standard deviations an element is from the mean. We often use z-scores in statistics to standardize different data points, facilitating easy comparison and calculation.

In our context, the z-score helps determine the sample size. Given a 95% confidence level, the corresponding z-score is 1.96. This score helps in creating the confidence interval, translating the given values into a standardized format:
  • A higher z-score corresponds to a higher confidence level, which increases the required sample size.
  • Z-scores are derived from standard normal distribution tables, simplifying the process of calculating confident intervals and sample sizes.
Understanding z-scores is essential because it allows us to standardize our methods, ensuring that statistical results are comparable across different studies and parameters.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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

Acrylic bone cement is sometimes used in hip and knee replacements to fix an artificial joint in place. The force required to break an acrylic bone cement bond was measured for six specimens under specified conditions, and the resulting mean and standard deviation were 306.09 Newtons and 41.97 Newtons, respectively. Assuming that it is reasonable to believe that breaking force under these conditions has a distribution that is approximately normal, estimate the mean breaking force for acrylic bone cement under the specified conditions using a \(95 \%\) confidence interval.

Each person in a random sample of 20 students at a particular university was asked whether he or she is registered to vote. The responses \((\mathrm{R}=\) registered, \(\mathrm{N}=\) not registered) are given here: R \(R N R N N R R R N R R R R R N R R R N\) Use these data to estimate \(p\), the proportion of all students at the university who are registered to vote.

The two intervals (114.4,115.6) and \((114.1,\) 115.9 ) are confidence intervals (computed using the same sample data) for \(\mu=\) true average resonance frequency (in hertz) for all tennis rackets of a certain type. a. What is the value of the sample mean resonance frequency? b. The confidence level for one of these intervals is \(90 \%\) and for the other it is \(99 \%\). Which is which, and how can you tell?

USA Today (October 14, 2002) reported that \(36 \%\) of adult drivers admit that they often or sometimes talk on a cell phone when driving. This estimate was based on data from a sample of 1004 adult drivers, and a bound on the error of estimation of \(3.1 \%\) was reported. Assuming a \(95 \%\) confidence level, do you agree with the reported bound on the error? Explain.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.