Motor insulation 2. Use the log data and summary statistics on motor insulation 2 in Table 5.2. The model for such data assumes that time to breakdown has a lognormal distribution at each temperature, that the (log) standard deviation is the same at each temperature, and that the (log) mean decreases with temperature. The following analyses assess whether the data are consistent with these assumptions.

(a) Plot the data (Table 5.1) from the three temperatures on the same lognormal paper, and compare the distributions.

(b) Calculate a pooled standard deviation from those from the three test temperatures.. How many degrees of freedom does it have?

(c) Calculate a separate two-sided 90% confidence interval for each ratio of pairs of (log) standard deviations for the three temperatures. Do any pairs differ statistically significantly?

(d) Calculate Bartlett's test statistic (5.14) for the (log) standard deviations of the three temperatures. How many degrees of freedom does the statistic have? Look up the 90 and 95% points of the corresponding

chi-square distribution. Do the (log) standard deviations differ statistically significantly? If so, how?

(e) How good is the chi-square approximation for (d)?

(f) Calculate simultaneous two-sided approximate 90% confidence limits for all ratios of pairs of the three standard deviations. Do any pairs differ wholly statistically significantly?

(8) Would you expect the conclusions from (d) and (f) to usually be the same? Why?

(h) Calculate two-sided 95% confidence limits for the (log) mean at 200°C. using the pooled standard deviation. Calculate the corresponding estimate and confidence limits for the lognormal median at 200°C.

(i) Calculate a separate two-sided 90% confidence interval for the difference of each pair of (log) means for the three temperatures, using the pooled (log) standard deviation. Do any pairs differ statistically significantly?

(j) Calculate the F statistic (5.18) for a one-way analysis of variance. How many degrees of freedom does it have in the numerator and in the denominator? Look up the 90 and 95% points of the corresponding F-distribution. Do the (log) means differ statistically significantly? If so, how?

(k) Calculate simultaneous two-sided approximate 90% confidence limits for all differences of pairs of the three (log) means. Do any pairs differ wholly statistically significantly?

(l) Would you expect the conclusions from (i) and (j) to usually be the same? Why?

(m) In view of the plots from (a), do you think that the analytic comparisons (b) through (k) are necessary? Helpful? What further information do the plots yield that the analytic methods do not?

Table 5.1