Confidence Intervals: Teacher's Guide 1.11 2008/06/09 15:14:53 GMT-5 2008/10/28 10:40:14.972 GMT-5 Susan Dean deansusan@deanza.edu Barbara Illowsky illowskybarbara@deanza.edu Susan Dean deansusan@deanza.edu Barbara Illowsky illowskybarbara@deanza.edu Connexions cnx@cnx.org elementary statistics 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 σ is known and when σ is not known (s is used as an estimate for σ).Book notation:CL = confidence level EBM = error bound for a mean EBP = error bound for a proportionThe student-t distribution in introduced in this chapter beginning with a little 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 σ is not known, replace σ with s, the sample standard deviation, and use the Student-t distribution. The shape of the curve depends on the parameter degrees of freedom (df). df=n−1 where n is the sample size. tdf designates the distribution. We use T as the random variable. Value is an average. The t-statistic (t-score)t = value-μ (σ n) The relationship between the confidence interval for a single mean (when σ and the confidence level can be shown in a picture as follows:

The α2 subscript indicates that the area to the right is α2.

Formulas for the error bounds:Single mean (known σ): EBM=zα 2⋅( σn) Single mean (unknown σ): EBM=tα 2⋅( sn) Binomial proportion: EBP=zα 2⋅p'q 'n where q=1− p' The confidence intervals have the form:Single mean (unknown or known σ ): (x- -EBM, x- +EBM) Binomial proportion: (p' -EBP, p' +EBP) 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.You want a confidence interval for a single mean where σ is not known. If you use the TI-83/84 series, enter the data into a list and then use the function TInterval, data option. C-level is 95. The confidence interval is (296.4, 334.2). This function also calculates the sample mean (315.3) and sample standard deviation (20.4). TInterval is found in STAT TESTS.If you want the students to use the formulas for a normal or for the Student-t confidence interval, you will need to use a table for the z-score or the t-score. The book does not have the tables but the Internet has several. Do a search on "z-score table" and "Student-t table."First, you need to calculate the sample mean and the sample standard deviation. x- =315.29 s=20.40 The confidence interval has the pattern : (x- -EBM, x- +EBM)The error bound formula is : EBM = tα2 ⋅(sn )CL = 0.95 so α=0.05. Therefore, α2=0.025.Using the Student-t table with df = 7-1=6 , t.025 = 2.45.

EBM= tα 2⋅(s n)= t0.25⋅ 20.407= 2.45⋅20.407 =18.89The confidence interval is (x- -EBM, x- +EBM) = (315.29 - 18.89, 315.29 +18.89) = (296.4, 334.2)We are 95% confident that the true average number of calories in a 4 ounce serving of french fries is between 196.4 and 334.2 calories. 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.You want a confidence interval for a single proportion. If you use the TI-83/84 series, use the function 1-PropZinterval. x=102 , n=450, C−level=80. The confidence interval is (.2077, .2590)If you want to use the formulas, first, you need to calculate the estimated proportion.p'=xn =102450= 0.23The confidence interval has the pattern (p' -EBP, p' +EBP).The error bound formula is EBP=zα 2⋅p'q 'n where q=1− p'CL=0.80 so α=0.20. Therefore, α2= 0.10.Using the normal table (find one on the Internet), z.10=1.28 . (Remind students that 0.10 is the area to the right. The area to the left is 0.90.)EBP=zα 2⋅p'q 'n =z.10⋅p'q 'n =z.10⋅ .23⋅.77n =1.28⋅ .23⋅.77450 =0.03 The confidence interval is : (p' -EBP, p' +EBP) = (0.23 - 0.03, 0.23 + 0.03) = (0.20, 0.26) We are 80% confident that the true proportion of families that have children on the swim team in any year is between 0.20 and 0.26. Assign PracticeAssign the Practice 1, Practice 2, and Practice 3 in class to be done in groups.Assign HomeworkAssign Homework. Suggested homework: 1, 5, 9, 13, 15, 17, 21, 23, 24 - 31.