Both a Type I and a Type II error? d. Neither a Type I nor a Type II error?

The data are analyzed by a standard program package SAS. The relevant output is shown here: a. What is the research hypothesis? b. What are the values of the t and t⬘ statistics? Why are they equal for this data set? c. What are the p-values for t and t⬘ statistics? Why are they different? d. Are the conclusions concerning the research hypothesis the same for the two tests if we use a ⫽ .05? e. Which test, t or t⬘, is more appropriate for this data set? Engin. 6.60 An industrial concern has experimented with several different mixtures of the four components—magnesium, sodium nitrate, strontium nitrate, and a binder—that comprise a rocket propellant. The company has found that two mixtures in particular give higher flare- illumination values than the others. Mixture 1 consists of a blend composed of the proportions .40, .10, .42, and .08, respectively, for the four components of the mixture; mixture 2 consists of a blend using the proportions .60, .27, .10, and .05. Twenty different blends 10 of each mixture are prepared and tested to obtain the flare-illumination values. These data appear here in units of 1,000 candles. Mixture 1 185 192 201 215 170 190 175 172 198 202 Mixture 2 221 210 215 202 204 196 225 230 214 217 a. Plot the sample data. Which tests could be used to compare the mean illumination values for the two mixtures? b. Give the level of significance of the test and interpret your findings. 6.61 Refer to Exercise 6.60. Instead of conducting a statistical test, use the sample data to answer the question, What is the difference in mean flare illumination for the two mixtures?

6.62 Refer to Exercise 6.60. Suppose we wish to test the research hypothesis that m

1 ⬍ m 2 for the two mixtures. Assume that the population distributions are normally distributed with a common s ⫽ 12. Determine the sample size required to obtain a test having a ⫽ .05 and bm d ⬍ .10 when m 2 ⫺ m 1 ⱖ 15. 6.63 Refer to the epilepsy study data from Chapter 3. An analysis of the data produced the fol- lowing computer output. The measured variable is the number of seizures after 8 weeks in the study for patients on the placebo and for those treated with the drug progabide. Two-Sample T-Test and Confidence Interval Two-sample T for Placebo vs Progabide N Mean StDev SE Mean Placebo 28 7.96 7.63 1.4 Progabide 31 6.7 11.3 2.0 95 CI for mu Placebo mu Progabide: 3.8, 6.3 T-Test mu Placebo mu Progabide vs : T 0.50 P 0.31 DF 57 Both use Pooled StDev 9.71 TEST PROCEDURE Variable: POTENCY SAMPLE N Mean Std Dev Std Error Variances T DF Prob T -------------------------------------------------- ---------------------------------- 1 10 10.37000000 0.32335052 0.10225241 Unequal 4.2368 16.6 0.0006 2 10 9.83000000 0.24060110 0.07608475 Equal 4.2368 18.0 0.0005 For HO: Variances are equal, F 1.81 DF 9,9 Prob F 0.3917 Boxplots of placebo and progabide means are indicated by solid circles Placebo Progabide

10 20

30 40 50 60 Number of seizures Boxplots of midsize and SUV damage amounts means are indicated by solid circles Midsize SUV 30 20 10 Damage amounts in hundreds of dollars

a. Do the data support the conjecture that progabide reduces the mean number of

seizures for epileptics? Use both a t test and the Wilcoxon test with a ⫽ .05. b. Which test appears to be most appropriate for this study? Why? c. Estimate the size of the differences in the mean number of seizures between the two groups. Bus. 6.64 Many people purchase sports utility vehicles SUVs because they think they are sturdier and hence safer than regular cars. However, preliminary data have indicated that the costs for re- pairs of SUVs are higher than for midsize cars when both vehicles are in an accident. A random sample of 8 new SUVs and 8 midsize cars are tested for front impact resistance. The amounts of damage in hundreds of dollars to the vehicles when crashed at 20 mph head on into a stationary barrier are recorded in the following table. Car 1 2 3 4 5 6 7 8 SUV 14.23 12.47 14.00 13.17 27.48 12.42 32.59 12.98 Midsize 11.97 11.42 13.27 9.87 10.12 10.36 12.65 25.23 a. Plot the data to determine whether the conditions required for the t procedures are valid. b. Do the data support the conjecture that the mean damage is greater for SUVs than for midsize vehicles? Use a ⫽ .05 with both the t test and Wilcoxon test. c. Which test appears to be the more appropriate procedure for this data set? d. Do you reach the same conclusions from both procedures? Why or why not? Two-Sample T-Test and Confidence Interval Two-sample T for Midsize vs SUV N Mean StDev SE Mean Midsize 8 13.11 5.05 1.8 SUV 8 17.42 7.93 2.8 95 CI for mu Midsize mu SUV: 11.4, 2.8 T-Test mu Midsize mu SUV vs : T 1.30 P 0.11 DF 14 Both use Pooled StDev 6.65