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Statistics 302H
Exam 2
April 5, 2006
Name: _______________________
Instructions: Always show how you set up the problem, in order to receive partial credit for a problem. On most problems, you don’t need to get the numerical answer, so long as you show how you would proceed to get the answer by inserting the appropriate numbers into the correct formula. You may use the book only for the tables in the back.
Background: The article below refers to a study published in the American Heart Journal (2006, Vol 151, p. 934-942) this week. A total of 1800 heart-bypass patients were divided into the following three groups.
Group Treatment Sample Size 1 People assigned to pray for patients, and patients knew it
2 People assigned to pray for patients, but patients did not know it
3 No one assigned to pray for patients 600 1800
The study looked for any complications within 30 days of surgery. Questions 1, 2 and 5 relate to this study.
- Let pk denote the probability of complication for patients that knew they were prayed for
(i.e. in group 1) and pdk the probability for those who didn’t know (i.e. in groups 2 and 3). The objective is to test whether a patient who knows about prayer would have a lower probability of complications than one who doesn’t know. Complete the following.
a) H 0 : Ha :
b) Test statistic and rejection region for α =.
c) The estimate of pk from the data is 0.59, and for pdk is 0.52. Calculate the test statistic.
d) Give the p -value
e) Give the conclusion about your statistical findings in the context of this problem.
- Twelve sets of identical twins were given psychological tests to measure their aggressiveness. The fundamental question of interest is whether the population mean of the first-born child differs from that of the second-born child. Their aggressiveness scores (arranged in order for first born) are:
Twin Set First-Born Second-Born Difference 1 68 64 4 2 70 65 5 3 71 77 - 4 71 80 - 5 72 72 0 6 77 76 1 7 77 65 12 8 86 88 - 9 87 72 15 10 88 81 7 11 91 90 1 12 91 96 - mean 79.1 77.2 1. st. deviation 8.9 10.4 7.
Some summary statistics are also given. Complete the following:
a) H 0 : Ha :
b) Test statistic
c) Rejection region α =.
d) Value of test statistic
e) p -value
f) Decision in context of the research hypothesis
- Consider the following data on the zodiac signs of 240 executives.
Aries 21 Libra 17 Taurus 18 Scorpio 20 Gemini 16 Sagittarius 18 Cancer 21 Capricorn 21 Leo 19 Aquarius 23 Virgo 18 Pisces 28
Test whether all the zodiac signs are equally likely among executives, i.e. whether each zodiac sign has probability of 1/12 among executives. Use α =.05.
a :
H
H
Test statistic (including degrees of freedom)
Rejection region.
Calculations. (Show how to set this up. You may use default statistic of 10.0 if you like.)
p -value
Decision
(Note: If you don’t solve this problem, note that Pisces is the sign for 28 of 240 executives. For less credit, test whether the true proportion for Pisces exceeds p = 1/12.)
c) The mean ‘cross clamp times’ were recorded for the three groups in Problem 2. The three means, in minutes, for the three groups are x 1 (^) = 65.7, x 2 (^) = 63.5, and x 3 =66.6.
H : ,
Suppose an ANOVA rejected the null hypothesis
0 μ 1 =^ μ^2 =μ 3
2
.
with within groups variance sw =1000.Use the Bonferroni procedure with overall level α = to test whether you can say that the population mean of the second group (with x 2 (^) = 63.5) is less than the mean of the third group (with x = 66.6).
test statistic p-value conclusion
