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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 for estimating \(\mu\) with known standard deviation \(\sigma\) is calculated using the formula \(E = z \cdot \frac{\sigma}{\sqrt{n}}\) where \(z\) is the z-score corresponding to the desired confidence level, \(\sigma\) is the known standard deviation, and \(n\) is the size of the sample.

Step by step solution

01

Understanding Margin of Error

Margin of error is a statistical concept that represents the amount of random sampling error in a survey's results. In this case, it used to estimate the population mean.
02

Knowing the Known Standard Deviation

Here, it is mentioned that the standard deviation \(\sigma\) is known. The standard deviation is a measure of how spread out numbers in the data are, and it's a key parameter when calculating the margin of error.
03

Understanding the Margin of Error Formula for Known Standard Deviation

For a known standard deviation \(\sigma\), the margin of error (E) formula is given by: \(E = z \cdot \frac{\sigma}{\sqrt{n}}\), where: \n- \(z\) is the z-score, which corresponds to the desired confidence level,\n- \(\sigma\) is known standard deviation of the population,\n- \(n\) is the size of the sample. N.B: The z-score depends on the desired confidence level. For instance, for a confidence level of 95%, the z-score is approximately 1.96.

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Most popular questions from this chapter

A company that produces detergents wants to estimate the mean amount of detergent in 64 -ounce jugs at a \(99 \%\) confidence level. The company knows that the standard deviation of the amounts of detergent in all such jugs is \(.20\) ounce. How large a sample should the company select so that the estimate is within \(.04\) ounce of the population mean?

For a population data set, \(\sigma=12.5\). How large a sample should be selected so that the margin of error of estimate for a \(99 \%\) confidence interval for \(\mu\) is \(2.50 ?\) 1\. How large a sample should be selected so that the margin of error of estimate for a \(96 \%\) confidence interval for \(\mu\) is \(3.20\) ?

Check if the sample size is large enough to use the normal distribution to make a confidence interval for \(p\) for each of the following cases. a. \(n=50 \quad\) and \(\quad \hat{p}=.25\) b. \(n=160\) and \(\hat{p}=.03\) c. \(n=400\) and \(\hat{p}=.65\) d. \(n=75 \quad\) and \(\quad \hat{p}=.06\)

Briefly explain how the width of a confidence interval decreases with a decrease in the confidence level. Give an example.

Check if the sample size is large enough to use the normal distribution to make a confidence interval for \(p\) for each of the following cases. a. \(n=80\) and \(\hat{p}=.85\) b. \(n=110 \quad\) and \(\quad \hat{p}=.98\) c. \(n=35\) and \(\hat{p}=.40\) d. \(n=200\) and \(\hat{p}=.08\)
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