"Unless statistics lie, he was morebrave than me: more blond than you."

E. E. Cummings

10.1 Estimating With Confidence (Pages 506 - 526)

OVERVIEW: Confidence intervals areimportant in statistics. This chapter provides information on howsuch intervals can be constructed from samples, and how to interpretsuch intervals.

C% confidence Interval for aparameter: An interval computed fromsample data by a method that has probability C% of producing aninterval containing the true value of the parameter.

A confidence interval for an unknownpopulation mean, m, calculated from a sample of size n with mean x(bar), hasthe form

x(bar) plus/minusz*(s/sqrt(n))

where z* is obtained from the normal distributiontable and s is the standard deviation of the population.

Example:
Suppose that the following scores represent a random sample from apopulation with a known standard deviation
s = 3.88. Find a 95% confidenceinterval for the mean of the population.

{85, 83, 91, 88, 88, 92, 81, 83, 85,83, 86, 84}

The mean of the sample is 85.75.

Note: Remember that the Central LimitTheorem of Statistics states that the means of sample means of aspecific size are normally distributed.

The 95% confidence interval for the mean of thepopulation from which the sample was chosen is
85.75 plus/minus 1.96(3.88/sqrt(12)). This is 85.75 plus/minus 2.20. The interval can be represented [83.55, 87.95].

Interpretation: If we took 20 random samples fromthe population and computed 95% confidence intervals for each sample,we would expect 19 of the 20 samples to contain the mean of thepopulation.

When we read the results of polls in newspapersand magazines produced by reliable polling organizations, we areoften provided with a margin of error. For instance, we might readthat 64% of those polled favor Proposition A, and the margin of erroris 3%. For those who are statistically literate, this determines a95% confidence interval [61%, 67%]. Confidence intervals forproportions is discussed in a latter chapter. The purpose inmentioning this here is simply to note that those who read magazinesand newspapers see (implied) confidence intervals published on adaily basis.

The margin of error formula has the formz*(s/sqrt(n)). In the example above, the margin of error is1.96(3.88/sqrt(12)) = 2.20. The margin of error actually determinesthe length of the confidence interval.

Suppose we want to produce a 95% confidenceinterval that is three units in length. That is, we want the marginof error to be no more than 1.5 units. We can do this be taking alarger sample. How large a sample would be needed? Well, we cananswer this question by solving the inequality

1.96(3.88/sqrt(n)) <1.5

Using basic algebra, one obtains

sqrt(n) > 1.96(3.88)/1.5 ==> n >[1.96(3.88)/1.5]2 = 25.70. Hence, we would need a sample size of 26or more to obtain a margin of error of 1.5 units.

The TI-83 can be used to obtain values of z* forconfidence intervals of various levels. Some examples:

95% confidence interval:invNorm(.975,0,1) = 1.96

99% confidence interval:invNorm(.995,0,1) = 2.58

90% confidence interval: invNorm(.95,0,1) = 1.64

80% confidence interval: invNorm(.90,0,1) = 1.28

It is strongly suggested that the section titledSome Cautions (pages 524 and 525) be read very carefully.