Copyright © 2014 Pearson Education. All rights reserved. 8.2-1 8.2 Estimating Population Means...

Copyright © 2014 Pearson Education. All rights reserved. 8.2-1

8.2 Estimating Population Means

LEARNING GOAL

Learn to estimate population means and compute the associated margins of error and confidence intervals.

Copyright © 2014 Pearson Education. All rights reserved. 8.2-2Slide 8.2- 2

Estimating a Population Mean: The Basics

When we have only a single sample, the sample mean is the best estimate of the population mean, μ.

However, we do not expect the sample mean to be equal to the population mean, because there is likely to be some sampling error. Therefore, in order to make an inference about the population mean, we need some way to describe how well we expect it to be represented by the sample mean.

The most common method for doing this is by way of confidence intervals.


A precise calculation shows that if the distribution of sample means is normal with a mean of μ, then 95% of all sample means lie within 1.96 standard deviations of the population mean; for our purposes in this book, we will approximate this as 2 standard deviations.

A confidence interval is a range of values likely to contain the true value of the population mean.


95% Confidence Interval for a Population MeanThe margin of error for the 95% confidence interval is

where s is the standard deviation of the sample.

We find the 95% confidence interval by adding and subtracting the margin of error from the sample mean. That is, the 95% confidence interval ranges

from (x – margin of error) to (x + margin of error)

We can write this confidence interval more formally as

x – E < μ < x + E

or more briefly as

x ± E

margin of error = E ≈2s n

x̄ x̄

x̄ x̄

x̄


Compute the margin of error and find the 95% confidence interval for the protein intake sample of n = 267 men, which has a sample mean of x = 77.0 grams and a sample standard deviation of s = 58.6 grams.

EXAMPLE 1 Computing the Margin of Error

x̄


Interpreting the Confidence Interval


EXAMPLE 2 Constructing a Confidence Interval

A study finds that the average time spent by eighth-graders watching television is 6.7 hours per week, with a margin of error of 0.4 hour (for 95% confidence). Construct and interpret the 95% confidence interval.


Choosing Sample Size

Choosing the Correct Sample SizeIn order to estimate the population mean with a specified margin of error of at most E, the size of the sample should be at least

where σ is the population standard deviation (often estimated by the sample standard deviation s).

22

n

E

22

sn

E

Solve the margin of error formula for n.E ≈ 2s / n


Solution:

EXAMPLE 3 Constructing a Confidence Interval

You want to study housing costs in the country by sampling recent house sales in various (representative) regions. Your goal is to provide a 95% confidence interval estimate of the housing cost. Previous studies suggest that the population standard deviation is about $7,200. What sample size (at a minimum) should be used to ensure that the sample mean is within

a. $500 of the true population mean?

b. $100 of the true population mean?

Copyright © 2014 Pearson Education. All rights reserved. 8.2-1 8.2 Estimating Population Means...

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