1. Following the explosion of the 25th space shuttle ﬂight, which was caused by an O-ring failure in one or more of the booster rockets, data from the previous 24 ﬂights were studied. The Temperature (F) at the time of launch and whether or not there was evidence of O-ring failures for each of the previous 24 shuttle ﬂights was determined for each ﬂight. A logistic regression relating the Failure of O-rings to the Temperature was obtained with the following result:
Estimate Std. Error z value Intercept 15.2954317 7.3280842 2.088 Temperature -0.2360002 0.1073615 2.198 a. What is the ﬁtted logistic regression equation for predicting the log(odds) of an O-ring failure?
b. If the temperature at launch was 63 F, what are the estimated odds of an O-ring failure?
c. What is the probability of an O-ring failure at 63 F?
d. What is the 95% conﬁdence interval estimate of the true slope β1 ?
2. A research project studied the physical properties of wood materials constructed by bonding together small ﬂakes of wood. The two factors considered were the size of the ﬂakes and species of tree. The sizes of the ﬂakes were S1: 0.15 inches by 2 inches and S2: 0.25 inches by 2 inches, and the species of tree used were aspen, birch, and maple. For each combination of ﬂake size and tree species, three samples of wood material were constructed. For each sample, the physical property measured was the tension modulus of elasticity in the direction perpendicular to the alignment of the ﬂakes, in pounds per square inch (psi). For these data, a two-way ANOVA was run and the partial ANOVA table is given below: df SS MSS F p value Source Size 157.44 .613 7256 Species Size∗ Species 41707 Error 12262 Total 17 206664 a. Fill in the blanks b. In the ANOVA table, the test for the main eﬀect of size has a P-value of 0.613. What does this indicate? i. The eﬀect of particle size probably varies considerably for the diﬀerent species. ii. For about 61.3%of the samples, there was a diﬀerence in the eﬀect of particle size. iii. For about 61.3% of the samples, there was no diﬀerence in the eﬀect of particle size. iv. None of the above.
c. Is the interaction eﬀect signiﬁcant?
3. A researcher is studying treatments for agoraphobia with panic disorder. The treatments are to be the drug Imipramine at the doses 1.5 mg per kg of body weight and 2.5 mg per kg of body weight. There will also be a control group given placebos. Thirty patients were randomly divided into three groups of 10 each. One group was assigned to the control and the other two groups were assigned to the two treatments. After 24 weeks on treatment, the subjects symptoms were evaluated through a battery of psychological tests, where high scores indicate a lessening of symptoms. Assume the data for the three groups are independent and the responses are approximately Normal. The means and standard deviations of the test scores for the three groups are given below: Group Mean St.dev Control 75.7 12.61 Dose = 1.5 84.1 18.441 Dose = 2.5 102.4 20.82 An ANOVA F test was run on the data. A partial ANOVA table is shown below: SS MSS F Source df Between 3727.8 Within 310.87 Total 29 a. Do the data show evidence of a violation of the assumption that the four populations have the same standard deviation?
b. What assumption is needed for an ANOVA can be check using a side-byside boxplots?
c. What is the value for the pooled standard deviation?
d. Find the value for the F statistics, its degrees of freedom, and explain what it is used for.
e. Suppose we are interested in the contrast that compares the high-dose group to the control group. What is a 90% conﬁdence interval for this contrast? Explain your ﬁndings.
f. Multiple-comparison procedures are going to be used to calculate simultaneous conﬁdence intervals for all pairwise comparisons using the Bonferroni method. For α = 0.05, the value of t** is 2.55. Explained how this value was found and ﬁnd a 95% conﬁdence...
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