1. 8.2 Estimating Population Means
LEARNING GOAL
Learn to estimate population means and compute the associated margins of error and confidence intervals.
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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.
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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.
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95% Confidence Interval for a Population Mean
The 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 x x
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EXAMPLE 1 Computing the Margin of Error
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. x
Solution: The sample size is n = 267 and the standard deviation for the sample is s = 58.6, so the margin of error is
E ≈ = = 7.2 2s n
2 × 58.6
267
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EXAMPLE 1 Computing the Margin of Error
Solution: (Cont.)
The sample mean is = 77.0 grams, so the 95% confidence interval extends approximately from 77.0 – 7.2 = 69.8 grams to = 84.2 grams. We write this result more formally as
or more simply as 77.0 ± 7.2 grams.
We are 95% confident that the population mean for protein intake of all American men is between 69.8 and 84.2 grams.
It is interesting to note that even the lower number in this confidence interval (69.8 grams) is greater than the recommended daily protein allowance for men of 55–60 grams, suggesting that actual protein consumption is significantly greater than recommended. x
69.8 grams < μ < 84.2 grams
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Interpreting the Confidence Interval
Figure 8.10 This figure illustrates the idea behind confidence intervals. The central vertical line represents the true population mean, μ. Each of the 20 horizontal lines represents the 95% confidence interval for a
particular sample, with the sample mean marked by the dot in the center of the confidence interval. With a 95% confidence interval, we expect that 95% of all samples will give a confidence interval that contains the population mean, as is the case in this figure, for 19 of the 20 confidence intervals do indeed contain the population mean. We expect that the population mean will not be within the confidence interval in 5% of the cases; here, 1 of the 20 confidence intervals (the sixth from the top) does not contain the population mean.
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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.
Solution: The best estimate of the population mean is the sample mean, = 6.7 hours.
We find the confidence interval by adding and subtracting the margin of error from the sample mean, so the interval extends from 6.7 – 0.4 = 6.3 hours to = 7.1 hours. x
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EXAMPLE 2 Constructing a Confidence Interval
Solution: (cont.)
We can therefore claim with 95% confidence that the average time spent watching television for the entire population of eighth-graders is between 6.3 and 7.1 hours, or
If 100 random samples of the same size were taken, we would expect the confidence intervals of 95 of those samples to contain the population mean.
6.3 grams < μ < 7.1 grams
Page Examples 3-5 on pages further illustrate constructing confidence intervals.
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Choosing Sample Size
Solve the margin of error formula for n.
E ≈ 2s / n E
Choosing the Correct Sample Size
In 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). E
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EXAMPLE 6 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?
Solution:
a. With E = $500 and σ estimated as $7,200, the minimum sample size that meets the requirements is
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EXAMPLE 6 Constructing a Confidence Interval
Solution:
(cont.) Because the sample size must be a whole number, we conclude that the sample should include at least 830 prices.
With E = $100 and σ = $7,200, the minimum sample size that meets the requirements is E
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Notice that to decrease the margin of error by a factor of 5 (from $500 to $100), we must increase the sample size by a factor of 25. That is why achieving greater accuracy generally comes with a high cost.
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TIME OUT TO THINK
If you decide you want a smaller margin of error for a confidence interval, should you increase or decrease the sample size? Explain.
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The End
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