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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 given by the formula \(E = Z \cdot \frac{\sigma}{\sqrt{n}}\), where \(Z\) is the z-score corresponding to the desired level of confidence, \(\sigma\) is the known population standard deviation and \(n\) is the size of the sample. Calculation involves substituting the correct values into this formula.

Step by step solution

01

Define Margin of Error

The margin of error of estimate for a population mean (\(\mu\)) when the population standard deviation (\(\sigma\)) is known is defined by a formula. The formula is derived from the concept of a z-score in statistics, which measures how many standard deviations an observation or datum is from the mean. The margin of error essentially states how 'off' we could be from the true population mean.
02

Introduce the Formula and Variables

The generally used formula for the margin of error when \(\sigma\) is known is: \(E = Z \cdot \frac{\sigma}{\sqrt{n}}\) where \(E\) represents the margin of error, \(Z\) represents the z-score, \(\sigma\) represents the population standard deviation, and \(n\) represents the size of the sample taken from the population. When calculating it, ensure that the value of Z corresponds to the required level of confidence.
03

The Calculation Process

To calculate the margin of error, you first identify or decide upon your desired confidence level, which then determines your 'Z' value (values from the Z-table). Secondly, you identify the known population standard deviation (\(\sigma\)) and the size of your sample (\(n\)). Substitute these values into the formula and calculate to find the margin of error.
04

Interpretation

After calculating, your result represents the range within which you can expect the true population mean to fall, with the given level of confidence. For example a margin of error of 2 implies, with the selected level of confidence, the estimate for the population mean will fall within 2 units of the true population mean.

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Most popular questions from this chapter

A hospital administration wants to estimate the mean time spent by patients waiting for treatment at the emergency room. The waiting times (in minutes) recorded for a random sample of 35 such patients are given below. \(\begin{array}{lrrrrrr}30 & 7 & 68 & 76 & 47 & 60 & 51 \\ 64 & 25 & 35 & 29 & 30 & 35 & 62 \\ 96 & 104 & 58 & 32 & 32 & 102 & 27 \\ 45 & 11 & 64 & 62 & 72 & 39 & 92 \\ 84 & 47 & 12 & 33 & 55 & 84 & 36\end{array}\) Construct a \(99 \%\) confidence interval for the corresponding population mean. Use the \(t\) distribution.

a. A sample of 1100 observations taken from a population produced a sample proportion of \(.32 .\) Make a \(90 \%\) confidence interval for \(p\). b. Another sample of 1100 observations taken from the same population produced a sample proportion of .36. Make a \(90 \%\) confidence interval for \(p\). c. A third sample of 1100 observations taken from the same population produced a sample proportion of .30. Make a \(90 \%\) confidence interval for \(p\). d. The true population proportion for this population is \(.34 .\) Which of the confidence intervals constructed in parts a through c cover this population proportion and which do not?

KidPix Entertainment 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 labor-hours 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?

A mail-order company promises its customers that the products ordered will be mailed within 72 hours after an order is placed. The quality control department at the company checks from time to time to see if this promise is fulfilled. Recently the quality control department took a sample of 50 orders and found that 35 of them were mailed within 72 hours of the placement of the orders. a. Construct a \(98 \%\) confidence interval for the percentage of all orders that are mailed within 72 hours of their placement. 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 alternative is the best?

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, \(\mu\).
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