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Chapter 8: Problem 8

What is the margin of error of estimate for \(\mu\) when \(\sigma\) is known? How is it calculated?

Short Answer

Expert verified
The margin of error of estimate for \(\mu\) when \(\sigma\) is known is calculated as \(E = Z \cdot \frac{\sigma}{\sqrt{n}}\), where \(Z\) is the Z-score associated with the desired confidence level, \(n\) is the sample size, and \(\sigma\) is the known population standard deviation. This gives the range in which we expect the true population mean to lie.

Step by step solution

01

Understanding Margin of Error

The margin of error is a statistic expressing the amount of random sampling error in the results of a survey. It is used to quantify the uncertainty in an estimate. The larger the margin of error, the less confidence one should have that the reported results are close to the 'true' figures; that is, the figures for the whole population.
02

Formula of Margin of Error

When the population standard deviation (\(\sigma\)) is known, the margin of error is calculated as: \(E = Z \cdot \frac{\sigma}{\sqrt{n}}\), where: \(Z\) is the Z-score, which varies depending on your desired confidence level, \(n\) is the sample size, and \(\sigma\) is the standard deviation of the population.
03

Explaining the Variables

\(\sigma\) is the population's standard deviation known from previous data or an educated guess. \(n\) is the sample size, or the number of data points you have. The Z-score, or \(Z\), is the number of standard deviations a given proportion is away from the mean for a normal distribution - in other words, it gives us a measure of how unlikely our result is assuming the null hypothesis is true. It is linked to the desired confidence level - commonly, for a confidence level of 95%, \(Z = 1.96\), while for a confidence level of 99%, \(Z = 2.58\).

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Key Concepts

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

Population Standard Deviation
The population standard deviation, symbolized as \( \sigma \), is a critical concept in statistics. It measures how much individuals in an entire population vary or deviate from the average or mean value.
  • Imagine you conducted a survey to measure people's heights in a city. If everyone is relatively the same height, the standard deviation will be low.
  • However, if there is a lot of variation in people's heights, the standard deviation will be high.
Knowing the population standard deviation allows researchers to calculate other statistical measures, such as the margin of error, with greater accuracy. It often requires a complete data set, or may rely on historical data and estimates when not every member of the population can be measured directly.
Z-score
The Z-score, often represented as \( Z \), is a statistical measure that assesses the standard deviations an element is from the mean of a dataset.
  • It tells you how typical or atypical your result is in relation to a normal distribution.
  • A Z-score of 0 suggests the value is exactly average, while a Z-score of 1.96 at a 95% confidence level means it's 1.96 standard deviations away from the mean.
Z-scores are crucial when calculating the margin of error, as they adjust depending on the desired confidence level. Higher confidence levels (like 99%) use higher Z-scores, increasing the margin of error, whereas lower confidence levels use smaller Z-scores.
Sample Size
The sample size, denoted by \( n \), refers to the number of observations or data points collected from a larger population. It significantly influences the reliability and accuracy of statistical estimates.
  • Larger sample sizes generally lead to more reliable results because they better represent the entire population.
  • In contrast, smaller sample sizes can lead to a higher margin of error due to greater variability and less representation of the whole.
When determining sample size, researchers must balance practicality and cost with the need for accuracy. The formula for margin of error illustrates this as \( E = Z \cdot \frac{\sigma}{\sqrt{n}} \), showing that as \( n \) increases, the margin of error \( E \) decreases.
Confidence Level
Confidence level is a statistical term that describes the percentage of certainty you have that your sample reflects the population. A higher confidence level means greater certainty that your interval estimate contains the true population parameter.
  • Common confidence levels include 90%, 95%, and 99%.
  • A 95% confidence level means that if you were to take 100 different samples and build an estimate interval using each sample, approximately 95 of the intervals would contain the true population parameter.
The confidence level affects the Z-score in statistical calculations. For instance, a 95% confidence level corresponds to a Z-score of 1.96, whereas a 99% confidence level uses a Z-score of 2.58. The higher the confidence level, the wider the margin of error, reflecting increased assurance in the estimate.

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Most popular questions from this chapter

A random sample of 20 acres gave a mean yield of wheat equal to \(41.2\) bushels per acre with a standard deviation of 3 bushels. Assuming that the yield of wheat per acre is normally distributed, construct a \(90 \%\) confidence interval for the population mean \(\mu\).

A random sample of 300 female members of health clubs in Los Angeles showed that they spend, on average, \(4.5\) hours per week doing physical exercise with a standard deviation of \(.75\) hour. Find a \(98 \%\) confidence interval for the population mean.

Determine the sample size for the estimation of the population proportion for the following, where \(\hat{p}\) is the sample proportion based on a preliminary sample. a. \(E=.025, \quad \hat{p}=.16, \quad\) confidence level \(=99 \%\) b. \(E=.05, \quad \hat{p}=.85, \quad\) confidence level \(=95 \%\) c. \(E=.015, \quad \hat{p}=.97, \quad\) confidence level \(=90 \%\)

In a random sample of 50 homeowners selected from a large suburban area, 19 said that they had serious problems with excessive noise from their neighbors. a. Make a \(99 \%\) confidence interval for the percentage of all homeowners in this suburban area who have such problems. b. Suppose the confidence interval obtained in part a is too wide. How can the width of this interval be reduced? Discuss all possible alternatives. Which option is best?

An entertainment company is in the planning stages of producing a new computer-animated movie for national release, so they need to determine the production time (labor-hours necessary) to produce the movie. The mean production time for a random sample of 14 big-screen computer-animated movies is found to be 53,550 labor-hours. Suppose that the population standard deviation is known to be 7462 laborhours and the distribution of production times is normal. a. Construct a \(98 \%\) confidence interval for the mean production time to produce a big-screen computer-animated movie. b. Explain why we need to make the confidence interval. Why is it not correct to say that the average production time needed to produce all big-screen computer-animated movies is 53,550 labor-hours?
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