Two Proportion Tests and Means

9.64
An auditor reviewed 25 oral surgery insurance claims from a particular surgical office, determining that the mean out-of pocket patient bill above the reimbursed amount was $275.66 with a standard deviation of $78.11. (a) At the 5 percent level of significance, does this sample prove a violation of the guideline that the average patient should pay no more than $250 out-of-pocket? State the hypothesis and decision rule. (b) Is this a close decision?

10.30
Note: For tests on two proportions, two means, or two variances it is a good idea to check you work using MIMITAB< Megstat, or the Learningstats, two-sample calculations in Unit 10.
In Dallas, some fire trucks were painted yellow (instead of red) to heighten their visibility. During a test period, the fleet of red fire trucks made 153,348 runs and had 20 accidents, while the fleet of yellow fire trucks made 135,035 runs and had 4 accidents. At &#945; =.01, did the yellow fire trucks have a significantly lower accident rate? (a) State the hypothesis. (b) State the decision rule and sketch it. (c) Find the sample proportions and z test statistics. (d) Make a decision. (e) Find the p-value and interpret it. (f) If statistically significant, do you think the difference is large enough to be important? If so, to whom, and why? (Is the normality assumption fulfilled? Explain.

Accident Rate for Dallas Fire trucks
Statistic Red Fire Trucks Yellow Fire Trucks
Number of accidents x1 =20 accidents x2 = 4 accidents
Number of fire runs n1 = 153,348 runs n2 = 135,035 runs

10.44
Does lovastatin (a cholesterol-lowering drug) reduce the risk of heart attack? In a Texas study researchers gave lovastatin to 2,325 people and an inactive substitute to 2,081 people (average age 58) After 5 years, 57 of the lovastatin group had suffered a heart attack, compared with 97 for the inactive pill. (a) State the appropriate hypothesis (b) Obtain a test statistic and p-value. Interpret the results at &#945; = .01 (c) Is normality assured? (d) Is the difference large enough to be important? (c) What else would medical researchers need to know before prescribing this drug widely? (Data from Science News 153 May 30, 1998. P 343.

10.46
To test the hypothesis that students who finish an exam first get better grader, Professor Hardtack kept track of the order in which papers were handed in. The first 25 papers showed mean score or 77.1 with a standard deviation of 19.6, while the last 24 papers handed in showed a mean score of 69.3 with a standard deviation 24.9. Is this a significant difference at &#945; = .05? (a) State the hypothesis for the right-tailed test. (b) Obtain a test statistic and p-value assuming equal variances. Interpret these results. (c) Is the difference in mean scores large enough to be important? (d) Is it reasonable to assume equal variances? (e) Cary out a formal test for equal variances at &#945; = .05, showing all steps clearly.

10.48
From her firm's computer telephone log, an executive found that the mean length of 64 telephone calls during July was 4.48 minutes with a standard deviation of 5.87 minutes. She vowed to make an effort to reduce to length of calls. The August phone log showed 48 telephone calls whose mean was 2.396 minutes with a standard deviation of 2.019 minutes. (a) State the hypothesis for a right-tailed test. (b) Obtain a test statistic and p-value assuming unequal variances. Interpret these results using &#945; = .01. (c) Why might the sample data not follow a normal, bell-shaped curve? If not, how might this affect your conclusions?