Assignment1_Spring2023
Assignment 1 – STAT 453/558
Due date: 20 January, 2023
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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:
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 Birth Order: 1 Birth Order: 2
1 5.73 6.08
2 5.80 6.22
3 8.42 7.99
4 6.84 7.44
5 6.43 6.48
6 8.76 7.99
7 6.32 6.32
8 7.62 7.60
9 6.59 6.03
10 7.67 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) The deflection temperature under load for two different formulations of ABS plastic
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 Formulation 2
206 193 192 177 176 198
188 207 210 197 185 188
205 185 194 206 200 189
187 189 178 201 197 203
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) Photoresist is a light-sensitive material applied to semiconductor wafers so that the
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 100 ºC
11.176 5.623
7.089 6.748
8.097 7.461
11.739 7.015
11.291 8.133
10.759 7.418
6.467 3.772
8.315 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. Use a = 0.05.
(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.
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