Exercise Sheet 7
MATH 0108
Gavin Shaddick
Question 1:
The flowing table gives the hours of relief for two different drugs, A and B, amongst
sufferers of a certain disease.
Patient
1
2
3
4
5
6
7
8
Hours of relief with
drug A
3.2
1.6
5.7
2.8
5.5
1.2
6.1
2.9
Hours of relief with
drug B
3.8
1.0
8.4
3.6
5.0
3.5
73
4.8
Mean
3.62
4.67
Relative advantage
of B in hours (x)
+0.6
-0.6
(a) In general, does drug B do better than drug A ?
(b) Perform an appropriate test to assess whether this is a significant improvement.
(c) Calculate a 99% confidence interval for the difference in hours of relief between
drug A and drug B
(d) What do you conclude ?
Question 2:
It was suspected that the stoats (Mustela erminea) on two neighbourhooding
islands were of different races. A number of animals were trapped and some body
measurements made. The tails, in mm, are given below
Island J
Island K
101
101
111
106
105
107
121
96
107
97
99
100
103
103
117
100
123
101
100
95
109
102
96
104
106
109
98
93
115
99
(a) Use this sample of data to perform hypothesis test whether there is a difference in
the length of tails between the stoats from the different islands (assuming that the
samples are independent)
(b) Calculate a 90% confidence interval for the true difference. Interpret your
findings.
Question 3:
In studies on moor matgrass (Nardus stricta) it was suspected that the grass, which grew
in well defined colonies, grew where the soil was locally deeper. To test this, two soil
depth measurements were made for each of 50 colonies, one within the colony and one
outside. The sum of the differences for the 50 pairs of measurements, d, was +637 cm.
The sum of the squares of the differences d2 was 14433.
(a) Examine the significance of the mean differences by using a paired t-test
(b) What do you conclude ?
(c) Could you perform an independent sample t-test using the data you have been
given ?
Question 4:
A breed of calf was feed a new type of food supplement, and to assess whether there were
any side effects, their heart rate was measured before and after a feed.
Calf
Before
After
Difference
1
2
105 79
109 87
+4
3
79
86
4
5
6
103 87 74
109 100 82
7
73
80
8
82
90
9
78
90
10
86
93
11
77
81
12
76
81
13
79
90
14
104
110
+6
(a) Use the sample of data given to assess whether there is an increase in the heart
rates of the calves after the feed. State your null and alternative hypothesis
carefully.
(b) Perform a hypothesis test, or other method, to assess whether the change is
significant.
(c) Which version of the t-test did you use, independent samples or paired ? State the
reasons for your choice.
Question 5:
The UK medical research council trial of hypertension in older adults reported its
findings in the BMJ in 1992. The trial recruited from general practices 4,396 patients
aged 65-74 whose systolic blood pressure was between 160 and 208 mmHg and
whose diastolic blood pressure was less than 115 mmHg. Patients were randomly
assigned to initial therapy with a diuretic or a beta-blocker or a matched placebo,
and then followed up for an average of 5.8 years. The main objective was to see if
anti-hypertensive treatment is effective
The main results of the study were as follows
Treatment
Diuretic
No of patients
Deaths
1081
134
Beta-blocker
1102
167
Placebo
2213
315
(a) Use an appropriate hypothesis test to assess whether active treatment (diuretic and
beta-blockers combined) reduces the risk of death
(b) Obtain the 95% confidence interval for the difference in proportion of deaths in
the active group (diuretic and beta-blockers combined) and the placebo group
(c) What do you conclude ?
(d) Perform an appropriate significance test to assess whether there is a difference in
the proportion of deaths in each of the three different treatments.
Question 6:
A random sample of 200 university staff were questioned and 34 suffered from back
pain. A random sample of 1000 students was taken, 43 said they suffered from back pain.
(a) Based on this sample, what is the probability that a randomly selected member of
staff will have back pain ?
(b) Perform a hypothesis test to test for a difference in the proportions of students and
staff reporting back pain
a. Perform a z-test of the difference in proportions
b. Perform a chi-test
c. Ensure that your answers from the two tests are consistent
Question 7:
The data in the following table is to be used to test for an association between smoking in
the household of the mother and low birth weight of her baby. The sample of 120
mothers.
Low birth weight
Smoking in
No
Yes
Total
household
Yes
43
5
48
No
71
1
72
Total
114
6
120
(a) Given this data, is there evidence of an association between smoking in the
household and low birth weight ?
a. Perform a z-test for the difference in proportions
b. Calculate a 95% confidence interval for the difference in proportions
c. Perform a Chi-squared test of association
(b) What do you conclude from your calculations ?
(c) Are all the tests you performed valid ? Carry out the chi-squared test with a
continuity correction, do your results differ widely?
Question 8:
Find these probabilities for a sample of 9 children, if 60% of them had German measles
by the time they were 12 years old.
(a) At least 5 have had German measles
(b) Exactly seven have had German measles
(c) More than three have had German measles
Question 9:
Find the mean, variance and standard deviation for each of these values of n and p when
the conditions for the binomial distribution are met
(a) n=100, p=0.75
(b) n=300, p=0.3
(c) n=1000, q=0.1
(d) n=50, p=2/5
(e) n=36, p=1/6
Question 10:
A survey found that 21% of Americans watch fireworks on TV (rather than going
outside) on July 4th. Find the mean, variance and standard deviation of the number of
individuals who watch fireworks on TV if a random sample of 1000 Americans is
selected.