1
2008-09
Homepage MATH11400
Statistics http://www.maths.bris.ac.uk/maejc/stats1/intro.html
Problem Sheet 9
Remember: when online, you can access the Statistics 1 data sets from an R console by typing
load(url("http://www.maths.bris.ac.uk/maejc/stats1/stats1.RData"))
Note: for each of the following questions your answer should include a statement of any model
assumptions, a brief description of your working at each stage of the test procedure including the
approximate p-value and the critical region for the test, and a summary of your conclusions. You
may nd it helpful to check your numerical calculations using the t.test() function in R.
*1. Eight athletes ran a 400 metre race at sea level and at a later meeting ran a 400 metre race
at high altitude. Their times in seconds are shown in the table below. Test at the null
hypothesis that race times are unaffected by altitude, against the alternative that race times
are greater at high altitude using a test procedure with signicance level = 0.05. The data
is contained in the Statistics 1 data sets runner.sea and runner.high respectively.
Runner
1
2
3
4
5
6
7
8
Sea Level
48.3 47.6 49.2 50.3 48.8 51.1 49.0 48.1
High Altitude 50.4 47.3 50.8 52.3 47.7 54.5 48.9 49.9
*2. To investigate the relative size of secretarial starting salaries in the public and private sector,
9 private sector posts and 10 public sector posts were chosen at random from jobs advertised on the web. The table below shows the advertised stating salaries (in 1,000). You
may assume that the population variances are the same in the private and public sectors.
Use the data to test at the 0.05-level the hypothesis that starting salaries are the same in the
two sectors against the alternative that private sector starting salaries are higher. The data
is contained in the Statistics 1 data sets private and public respectively.
Private sector 12.1 13.4 11.3 10.6 9.7 12.5 9.6 13.6 11.2
Public sector
9.3 8.5 8.2 13.1 8.8 11.9 10.1 9.8 12.2 10.4
3. The design of controls can have a large effect on their ease of use. An investigator asked
20 right-handed subjects to to turn a screw-action indicator knob with their right hand. The
subjects were assigned at random to one of two identical instruments one with a righhanded thread where the knob turned clockwise and one with a left-handed thread where
the knob turned anti-clockwise. The table below gives the times required (in seconds) to
move the indicator a xed distance. Use the data to test whether or not there is signicant
difference in the time average required between the two types of knob, using a test procedure with signicance level = 0.05. The data is contained in the Statistics 1 data sets
thread.right and thread.left respectively.
Subject
1
2
3
4
5
6
7
8
9
Right-thread 113 105 130 101 138 118 87 116 96
Left-thread
137 105 133 108 115 170 103 145 107
10
122
84
Subject
11 12 13 14 15
Right-thread 103 116 107 118 111
Left-thread
148 147 87 166 123
20
88
123
1
16 17
89 100
93 116
18 19
89 85
78 101
4. A car manufacturer claims that the level of nitrous oxide emissions from its new engine is
lower than from its old engine. Evaluation of eight engines of the old type and nine of the
new type give the results for nitrous oxide emissions in parts per million in the table below.
Assuming the change of engine has not affected the variance of the distribution of emission
levels, test the hypothesis that there is no difference in the mean level of emissions of the
two models against an appropriate alternative. The data is contained in the Statistics 1 data
sets no2.new and no2.old respectively.
New model
Old model
1.24 0.79 0.95 0.51 1.47 1.03 1.33 1.00 1.05
1.38 1.41 1.04 0.88 1.12 1.81 1.29 1.18
5. A study was conducted of 10 households to see if alerting them to high usage rates of electricity reduced their actual consumption. A small monitor was installed in each household,
which activated a red ashing light whenever the current rate of usage exceeded a preset threshold. The monthly usage (in kilowatt-hours) before and after installation of the
monitor is given below. Test at the 0.05-level whether the monitor is effective at reducing
electrical consumption. The data is contained in the Statistics 1 data sets kwh.before
and kwh.after respectively.
Household
Before
After
1
2
3
4
5
6
7
8
9
10
940 1370 1030 2030 1540 2300 1800 910 640 1200
900 1230 1060 2100 1250 2200 1820 900 630 1110
6. In a study to examine whether increasing the amount of calcium in the diet reduced blood
pressure, a group of 10 men were given a calcium supplement in their diet for 12 weeks,
and a control group of 11 men received a placebo (a pill that appeared identical, but contained no active substance). The table below shows the relative change (in mm of mercury) in blood pressure over the 12 week period (before - after) for each subject. What do
you conclude from the results shown? The data is contained in the Statistics 1 data sets
bp.calcium and bp.placebo respectively.
Calcium group
Placebo group
7
4
1 12
18
1
17
35
3 3 5
2
1
5
10 11 2
2
11 1
3