Assignment 1 – STAT 453/558 Due date: 20 January, 2023
1) Suppose that we are testing H0: μ1 = μ2 versus H1: μ1 > μ2 with a sample size of n1 = n2 = 10. Both sample variances are unknown but assumed equal. Using R, find p-values for the following observed values of the test statistics:
(a) t0 = 2.45
(b) t0 = -3.60
(c) t0 = 1.96
(d) t0 = -2.19
(e) Repeat (a)-(d) for the case when the alternative hypothesis is two-sided.
2) The time to repair an electronic instrument is a normally distributed random variable measured in hours. The repair time for 16 such instruments chosen at random are as follows:
Hours
159 280 101 212 224 379 179 264 222 362 168 250 149 260 485 170
(a) Is the normality assumption appropriated? Justify.
(b) You wish to know if the mean repair time exceeds 225 hours. Set up appropriate
hypotheses for investigating this issue.
(c) Using R, test the hypotheses you formulated in part (b). What are your conclusions? Use a = 0.01.
(d) By hand, construct a 95 percent confidence interval on mean repair time to test your hypothesis in (b). Show your work.
3) An article in the journal of Neurology (1998, Vol. 50, pp.1246-1252) observed that the monozygotic twins share numerous physical, psychological and pathological traits. The investigators measured an intelligence score of 10 pairs of twins. The data are obtained as follows:
Twin pair
1 2 3 4 5 6 7 8 9 10
Birth Order: 1
5.73 5.80 8.42 6.84 6.43 8.76 6.32 7.62 6.59 7.67
Birth Order: 2
6.08 6.22 7.99 7.44 6.48 7.99 6.32 7.60 6.03 7.52
(a) Is the assumption that the difference in score is normally distributed reasonable?
(b) Using R, find a 95% confidence interval on the difference in the mean score. Is there any evidence that mean score depends on birth order?
(c) Test an appropriate set of hypotheses indicating that the mean score does not depend on birth order.
4) ThedeflectiontemperatureunderloadfortwodifferentformulationsofABSplastic pipe is being studied. Two samples of 12 observations each are prepared using each formulation, and the deflection temperatures (in °F) are reported below:
Formulation 1
206 193 192 188 207 210 205 185 194 187 189 178
Formulation 2
177 176 198 197 185 188 206 200 189 201 197 203
Do the data support the claim that the mean deflection temperature under load for formulation 2 exceeds that of formulation 1? (a) Use a = 0.05 to perform a complete analysis in R, including normality check and the appropriate test. Use the rejection region method to test your hypothesis. (b) Does the confidence interval support your conclusion on part a? Justify.
5) Photoresistisalight-sensitivematerialappliedtosemiconductorwaferssothatthe circuit pattern can be imaged on to the wafer. After application, the coated wafers are baked to remove the solvent in the photoresist mixture and to harden the resist. Here are measurements of photoresist thickness (in kÅ) for eight wafers baked at two different temperatures. Assume that all of the 16 runs were made in random order. Note: a wafer cannot be baked twice.
95 oC 11.176 7.089 8.097 11.739 11.291 10.759 6.467 8.315
100 oC 5.623 6.748 7.461 7.015 8.133 7.418 3.772 8.963
(a) Is there evidence to support the claim that the higher baking temperature results in wafers with a lower mean photoresist thickness? Use a = 0.05 and justify your answer. (b)Find a 95% confidence interval on the difference in means. Provide a practical interpretation of this interval.
6) The following are the burning times (in minutes) of chemical flares of two different formulations. The design engineers are interested in both the means and variance of the burning times.
Type 1 Type 2 65 82 64 56 81 67 71 69 57 59 83 74 66 75 59 82 82 70 65 79
(a) Test the hypotheses that the two variances are equal.
(b) Using the results of (a), test the hypotheses that the mean burning times are equal. Use a = 0.01. What is the p-value for this test?
(c) Discuss the role of the normality assumption in this problem. Check the assumption of normality for both types of flares.
Use a = 0.05.