<< Chapter < Page | Chapter >> Page > |
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 is known and when is not known ( is used as an estimate for ).
The student-t distribution in introduced in this chapter beginning with a little history:
If you sample from a normal distribution in which is not known, replace with , the sample standard deviation, and use the Student-t distribution. The shape of the curve depends on the parameter degrees of freedom ( ). where is the sample size.
The relationship between the confidence interval for a single mean (when and the confidence level can be shown in a picture as follows:
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
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.
The confidence interval has the pattern :
The error bound formula is :
so . Therefore, .
Using the Student-t table with , .
The confidence interval is = (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. , , . The confidence interval is (.2077, .2590)
If you want to use the formulas, first, you need to calculate the estimated proportion.
The confidence interval has the pattern .
The error bound formula is where
so . Therefore, .
Using the normal table (find one on the Internet), . (Remind students that 0.10 is the area to the right. The area to the left is 0.90.)
The confidence interval is : = (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.
Notification Switch
Would you like to follow the 'Collaborative statistics teacher's guide' conversation and receive update notifications?