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Ch. 8: Confidence Intervals

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

Summary: This module is the complementary teacher's guide for the Confidence Intervals chapter of the Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

Confidence intervals can be difficult for students. This chapter discusses confidence intervals for a single mean and for a single proportion. In this course, we do not deal with confidence intervals for two means or two proportions. For a single mean, confidence intervals are calculated when σσ is known and when σσ is not known (ss is used as an estimate for σσ).

Book notation:

  • CL = confidence level
  • EBM = error bound for a mean
  • EBP = error bound for a proportion

The student-t distribution in introduced in this chapter beginning with a little history:

history:

William Gossett derived the t-distribution in 1908. He needed a method for dealing with small samples (less than 30) in his research on temperature at the Guinness Brewery. Legend has it that the name Student-t comes from the fact that Gossett wrote a paper about the t-distribution and signed the paper Student because he was too modest to use his own name.

If you sample from a normal distribution in which σσ is not known, replace σσ with ss, the sample standard deviation, and use the Student-t distribution. The shape of the curve depends on the parameter degrees of freedom (dfdf). df=n−1df=n−1 where nn is the sample size.

Notation:

tdftdf designates the distribution. We use TT as the random variable. Value is an average.

The t-statistic (t-score)

t = value-μ (σ n) t = value-μ (σ n)

The relationship between the confidence interval for a single mean (when σσ and the confidence level can be shown in a picture as follows:

Figure 1: The α2α2 subscript indicates that the area to the right is α2α2.
Graph illustrating the distribution of a confidence interval when the mean value is known and the confidence level.

Formulas for the error bounds:

  • Single mean (known σσ): EBM=zα 2⋅( σn)EBM=zα 2⋅(σn)
  • Single mean (unknown σσ): EBM=tα 2⋅( sn)EBM=tα 2⋅(sn)
  • Binomial proportion: EBP=zα 2⋅p'q 'nEBP=zα 2⋅p'q 'n where q=1− p'q=1−p'

The confidence intervals have the form:

  • Single mean (unknown or known σσ ): (x- -EBM, x- +EBM)(x--EBM,x-+EBM)
  • Binomial proportion: (p' -EBP, p' +EBP)(p'-EBP,p'+EBP)

Example 1

Problem 1

The number of calories in fast food is always of interest. A survey was taken from 7 fast food restaurants concerning the number of calories in 4 ounces of french fries. The data is 296, 329, 306, 324, 292, 310, 350. Construct a 95% confidence interval for the true average number of calories in a 4 ounce serving of french fries.

[ Show Solution ]

Example 2

Problem 1

At a local cabana club, 102 of the 450 families who are members have children who swam on the swim team in 1995. Construct an 80% confidence interval for the true proportion of families with children who swim on the swim team in any year.

[ Show Solution ]

Assign Practice

Assign the Practice 1, Practice 2, and Practice 3 in class to be done in groups.

Assign Homework

Assign Homework. Suggested homework: 1, 5, 9, 13, 15, 17, 21, 23, 24 - 31.

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