For the 23 space shuttle flights that occurred before the Challenger mission in 1986, the table below shows the temperature (F) at the time of the flight and whether at least one of the six primary O-rings suffered thermal distress (1 = yes, 0 = no). The attached R output shows the use of various models for analyzing these data. Answer the question

1.). For the logistic regression model using temperature as a predictor for the probability of thermal distress, calculate the estimated probability of thermal distress at 31, the temperature at the time of the Challenger flight.

2) At the temperature at which the estimated probability equals 0.5, give a linear approximation for the change in the estimated probability per degree increase in temperature.

3) Interpret the estimated effect of temperature on the odds of thermal distress.

4). Test the hypothesis that temperature has no effect, using the likelihood-ratio test. Interpret the results.

5). Suppose we treat the (0, 1) response as if it has a normal distribution and fit a linear model for the probability. Report the prediction equation and find the estimated probability of thermal distress at 31. Comment on the suitability of this model.

6) Suppose we also want to include the month during which the launch occurred (January, February, etc.) in the model. Show how you could add indicator variables to the model to allow this? Explain how to interpret the coefficients of the indicator variable.

7) the previous part. Explain how you could further generalize the model to allow interaction between temperature and the month of the launch and explain how you could conduct a test to investigate whether you need the interaction term.

Ft Temperature TD Ft Temperature TD 66 13 67 70 14 53 69 15 67 68 16 75 17 70 18 81 - O - OOOOOO - O O - - - OOO OOO - C 19 76 70 20 79 57 21 75 63 22 76 70 23 58 78 Note: Ft = flight no., TD = thermal distress (1 = yes, 0 = no). Source: Data based on Table 1 in S. R. Dalal, E. B. Fowlkes and B. Hoadley, J. Am. Statist. Assoc., 84: 945-957, 1989. Reprinted with the permission of the American Statistical Association.Model 1: Call : glm(formula = TD - Temperature, family = binomial(link = logit)) Deviance Residuals: Min 10 Median 30 Max -1. 0611 -0.7613 -0.3783 0 . 4524 2.2175 Coefficients: Estimate Std. Error z value Pr(>Izl) (Intercept) 15.0429 7.3786 2.039 0. 0415 * Temperature -0.2322 0. 1082 -2.145 0. 0320 * Signif. codes: 0 '* ** ' 0. 001 '** ' 0.01 *' 0.05 '.' 0.1 ' ' 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 28.267 on 22 degrees of freedom Residual deviance: 20.315 on 21 degrees of freedom AIC: 24.315 Number of Fisher Scoring iterations: 5Model 2: Call: glm(formula = TD ~ Temperature, family = gaussian) Deviance Residuals: Min 10 Median 30 Max -0. 43762 -0. 30679 -0.06381 0. 17452 0. 89881 Coefficients: Estimate Std. Error t value Pr(>Itl) (Intercept) 2.90476 0. 84208 3.450 0.00240 * * Temperature -0.03738 0. 01205 -3. 103 0. 00538 * * Signif. codes: 0 '***' 0. 001 '**' 0.01 '*' 0.05 '. ' 0.1 ' ' 1 (Dispersion parameter for gaussian family taken to be 0. 1589796) Null deviance: 4.8696 on 22 degrees of freedom Residual deviance: 3.3386 on 21 degrees of freedom AIC: 26.882 Number of Fisher Scoring iterations: 2