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Module 14: Confidence Intervals

This module explores the development and interpretation of confidence intervals, with a focus on confidence intervals for the population mean, based on the sample mean.

Reviewed 05 May 05 / Module 14

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Some Questions

The following questions reference a normal distribution with a mean μ = 150 lbs, a variance σ2 = 100 lbs2, and a standard deviation σ = 10 lbs or N(150, 10).

1.   When centered about μ = 150 lbs, what proportion of the total distribution does an interval of length 10

lbs cover?

2. How many standard deviations long must an interval be in order to cover the middle 95% of the

distribution?

3.   From μ - (??) standard deviations to μ + (??) standard deviations covers (??)% of the distribution?

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All these situations require that the population mean μ be known and that it be placed in the center of these intervals.

In general, we have a sample mean, , and are using it as a guess or estimate for the population mean μ. Hence, it would be useful if we could have an interval based on , that is, with in the middle.

Such an interval would help us to understand better our point estimate, the sample mean , which is an estimate of the population mean μ.

x

xx

x

The Situation

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Confidence Interval for Population Mean

The uncertainty, ??, is based on:

1. The variability of , that is, the standard error

of the mean,

2. Probability, e.g., normal distribution

3. Confidence desired, i.e., and 1- .

x

x

n

We can write the formal interval

C[ - ?? + ??] = 1- .x x

x??x ??x

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1.96 1.96 0.95

?? ?? 1

In terms of standard errors of the mean,

? ? 1

If = 0.05, then 1- = 0.95 so that

C x xn n

C x x

C x xn n

Confidence Interval for Population Mean

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1.96 1.96 0.95C x xn n

For a 95% Confidence Interval, the following

 

becomes

1 2 1 2 1C x z x zn n

95% Confidence Interval for Population Mean

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Confidence Interval for the Population Mean using the Normal Distribution

Lower limit

1 2 1 2 1C x z x zn n

Upper limit

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5, 153.0, 10

10 101.96 1.96 0.95

5 5

becomes

10 10153.0 1.96 153.0 1.96 0.95

5 5

n x

C x x

C

For = 0.05 for the first of the random samples of size n = 5 from the population of body weights, we had

Example 1

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or

C 153.0-1.96 4.47 153.0 1.96 4.47 0.95

or

C 153.0-8.76 153.0 8.76 0.95

or

C 144.24 161.76 0.95

Example 1 (contd.)

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95% Confidence Intervals for samples n = 5

0.975 0.975C[ - ] = 95%n n

x z x z

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For = 0.05 for the first of the random samples of size n = 20 from the population of body weights, we had

20, 151.6, 10

10 101.96 1.96 0.95

20 20

becomes

10 10151.6 1.96 151.6 1.96 0.95

20 20

n x

C x x

C

Example 2

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or

C 151.6-1.96 2.23 151.6 1.96 2.23 0.95

or

C 151.6-4.37 151.6 4.37 0.95

or

C 147.2 155.9 0.95

Example 2 (contd.)

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95% Confidence Intervals for samples n = 20

0.975 0.975C[ - ] = 95%n n

x z x z

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95% Confidence Intervals for samples n = 50

0.975 0.975C[ - ] = 95%n n

x z x z

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For = 0.01 for the first of the random samples of size n = 5 from the population of body weights, we had

0.9950.995 0.995

5, 153.0, 10

- + ; = 2.576

10 102.576 2.576 0.99

5 5

becomes

10 10153.0 2.576 153.0 2.576 0.99

5 5

n x

C x z x zn n

C x x

C

z

Example 3

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or

C 153.0-2.576 4.47 153.0 2.576 4.47 0.99

or

C 153.0-11.51 153.0 11.51 0.99

or

C 141.5 163.1 0.99

Example 3 (contd.)

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For = 0.01 for the first of the random samples of size n = 20 from the population of body weights, we had

0.99520, 151.6, 10 ; = 2.576

10 102.576 2.576 0.99

20 20

becomes

10 10151.6 2.576 151.6 2.576 0.99

20 20

n x z

C x x

C

Example 4

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or

C 151.6-2.576 2.23 151.6 2.576 2.23 0.99

or

C 151.6-5.74 151.6 5.74 0.99

or

C 145.9 157.3 0.99

Example 4 (contd.)

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Confidence intervals for the population mean , are an estimation procedure with reasonable bounds about the sample mean ,

In general, the closer the bounds are to the point estimate, , the better the point estimate.

The bounds are constructed in a manner that takes into account the variability of the point estimate.

The bounds are also based on an appropriate probability distribution so that some reasonable probability statements can be made.

x

x

Confidence Intervals

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Our confidence is in the process we used to generate a specific confidence interval and not in the specific interval itself.

In general, we construct such intervals so that, should we repeat the process a large number of times, then 95%, for a 95% confidence interval, of such intervals should contain the population parameter being estimated by the point estimate and the confidence interval.

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The specific interval we compute in any given situation may or may not contain the population parameter.

The only way for us to be sure that the population parameter is within the bounds of the confidence interval is to know the true value for this parameter.

Obviously, if we knew the true value, we would not bother to go through the process of guessing at the truth with estimates.

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In the following example (Table 1, Elola et al, AJPH, 1995, 85: p1398), the mean and SD for health expenditures per capita (US$) for the n = 7 countries with Social Security Systems are given.

Questions:

1. What is your best guess as to the population mean for the population from which this sample was selected?

AJPH Example--Mean, SD, Variance, CI 

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2. What is the sample variance for the individual measures of health expenditures in this sample of size n = 7?

3. Construct a 95% confidence interval for this population mean. For this calculation, assume that the population variance is $22,500 (dollars2) and that the population standard deviation is $150.

4. What is the variance and standard deviation for the population of means of all possible samples of size

n = 7 for this situation?

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Source: AJPH, October 1995, 85:1398

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Suppose we have a random sample of n = 25 measurements of chest circumference from a population of newborns with = 0.7 in. The sample mean is = 12.6 in.

 A 95% confidence interval for is:

x

1.96 1.96 0.95

0.7 0.712.6 1.96 12.6 1.96 0.95

5 5

12.6 0.27 12.6 0.27 0.95

12.33 12.87 0.95

C x xn n

C

C

C

Another Example

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12.6 13.0 ?

12.6 13.0 ?

Is it possible that this sample with xcame from a population with

Is it likely that this sample with xcame from a population with

Questions