51. PERFORMING CONFIDENCE INTERVAL FOR TWO

CONFIDENCE INTERVAL FOR 2-SAMPLE MEANS

Case 1: calcium and blood

pressure Does increasing the amount of calcium in our diet reduce blood pressure? Examination of a large sample of people revealed a relationship between calcium intake and blood pressure. The relationship was strongest for black men. Such observational studies do not establish causation. Researchers therefore designed a randomized comparative experiment.

The subjects in part of the experiment were 21 healthy black men. A randomly chosen group of 10 of the men received a calcium supplement for 12 weeks. The control group of 11 men received a placebo pill that looked identical. The experiment was double-blind. The response variable is the decrease in systolic (top number) blood pressure for a subject after 12 weeks, in millimeters of mercury. An increase appears as a negative response.

Take Group 1 to be the calcium group and Group 2 the placebo group. Here are the data for the 10 men in Group 1 (calcium),

and for the 11 men in Group 2 (placebo),

From the data, calculate the summary statistics:

Confidence interval For a 90% confidence interval, Table C shows that the t(9) critical value is t* = 1.833. We are 90% confident that the mean advantage of calcium over a placebo, μ1 = μ2, lies in the interval

Warm-up Teaching reading, I An educator believes that new reading activities in the classroom will help elementary school pupils improve their reading ability. She arranges for a third-grade class of 21 students to follow these activities for an 8week period. A control classroom of 23 third-graders follows the same curriculum without the activities. At the end of the 8 weeks, all students are given the Degree of Reading Power (DRP) test, which measures the aspects of reading ability that the treatment is designed to improve. Here are the data:

(a) Is there good evidence that the new activities improve the mean DRP score? Carry out a test and report your conclusions. (b) Construct and interpret a 95% confidence interval for the difference in mean DRP scores.

Answers: (a) H0: μt = μc vs. Ha: μt > μc. t = 2.311, 0.01 < P–value < 0.02 with df = 20 (TI calculator gives P–value = 0.0132 with df = 37.86). (b) (0.97, 18.94) with df = 20; (1.233, 18.68) on TI calculator with df = 37.86. With 95% confidence, we estimate the mean improvement in reading ability using the new reading activities compared to not using them over an 8–week period to be between 1.23 and 18.68 points.