Question 1. Carpal tunnel syndrome is a painful wrist condition that can be treated with surgery or less invasively with wrist splints. In a study of 180 patients with the condition, half had surgery and half used wrist splints. In the surgery group, 70 patients showed improvement after three months while 42 of those who used wrist splints improved. Is surgery more effective than the use of wrist splints for improving symptoms of the condition?
a) Write appropriate hypotheses.
b) Test the hypothesis, find the P-value and state your conclusion. Use α = 0.025.
c) Create a 95% confidence interval for the difference in improvement rates, and interpret your interval.
d) Comment on your interval in relation to your conclusion from b).
Question 2. During an angiogram, heart problems can be examined via a small tube (a catheter) threaded into the heart from a vein in the patient's leg. It is important that the company that manufactures the catheters maintains a diameter of 2.00 mm. A random sample of 30 catheters is taken and the average diameter is 2.03 mm with standard deviation 0.05 mm.
a) Create a 95% confidence interval for the mean diameter of catheters produced by the company.
b) Explain in context what your interval means.
c) Perform a hypothesis test to find out if the mean diameter of the catheters is significantly different to the required 2.00 mm. Use a significance level of 5% and give your conclusion.
Question 3. A researcher wanted to see whether there is a significant difference in resting pulse rates for men and women. The data collected are summarised below.
| Sex |
|
Male |
Female |
| Count |
28 |
24 |
| Mean |
72.75 |
72.625 |
| Median |
73 |
73 |
| Studev |
5.37225 |
7.69987 |
| Range |
20 |
29 |
| IQR |
9 |
12.5 |
a) Are resting pulse rates for men and women significantly different? Carry out a hypothesis test. Use a significance level of 5% and use df = n1+n2-2.
b) Create a 95% confidence interval for the difference in mean pulse rates, and interpret the interval.
c) Does the confidence interval confirm your answer to a)? Explain.
Question 4. A researcher wishes to examine the effect of group therapy on weight loss for overweight people. A sample of 10 overweight people are randomly selected and their weights (in kg) recorded before and after the therapy program.
|
Subject
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
|
Initial weight
|
72
|
85
|
68
|
81
|
92
|
76
|
84
|
102
|
69
|
74
|
|
Final weight
|
67
|
88
|
63
|
72
|
94
|
73
|
89
|
87
|
75
|
68
|
a) Is there evidence that the group therapy helped with weight loss? Carry out a hypothesis test (by hand or with SPSS) using α = 0.05 and write a conclusion.
b) Calculate a 90% confidence interval estimate for the average weight loss. Based on your hypothesis test result, did you expect to find 0 in the interval? Explain.
Question 5. Smarties (sugar coated chocolate confectionary) come in 8 colours - green, yellow, red, orange, pink, purple, blue and brown. You buy a bag containing 120 smarties to investigate the distribution of colours, and count 12 green, 14 yellow, 17 red, 15 orange, 16 pink, 17 purple, 11 blue and 18 brown smarties.
a) If Smarties are packaged in equal proportions, how many of each colour would you expect in the bag?
b) To see if your results are unusual, will you perform a goodness-of-fit test or a test of independence?
c) State your hypotheses.
d) Check the conditions.
e) How many degrees of freedom are there?
f) Find χχ2 and the P-value.
g) State your conclusion.
Question 6. The following table shows data on randomly selected crime victims. Are crime and the relationship of the criminal to the victim independent?
|
|
Homicide
|
Robbery
|
assault
|
|
Criminal was a stranger
|
12
|
379
|
727
|
|
Criminal was known to victim
|
39
|
106
|
642
|
a) Write appropriate hypotheses.
b) How many degrees of freedom are there?
c) Find χχ2 and the P-value.
d) State your conclusion and analysis.
Question 7. Wild bears were caught and anesthetised so that various measurements could be made. In particular, the usefulness of a bear's chest circumference to predict its weight was of interest. A random sample of 10 bears was used, with the chest and weight measurements shown below, as well as the linear regression analysis.
| Bear |
Chest |
weight |
|
(inches) |
(pounds) |
| 1 |
26 |
80 |
| 2 |
45 |
344 |
| 3 |
54 |
416 |
| 4 |
49 |
348 |
| 5 |
35 |
166 |
| 6 |
41 |
220 |
| 7 |
41 |
262 |
| 8 |
49 |
360 |
| 9 |
38 |
204 |
| 10 |
31 |
144 |
|
|
|
Unstandardized
Coefficients
|
Standardized
Coefficients
|
|
|
|
Model
|
|
B |
Std. Error
|
Beta |
t
|
Sig. |
|
1 (Constant
|
-251.948
|
33.814
|
|
-7.451
|
.000
|
|
CHEST
|
12.380
|
.810
|
.983
|
15.277
|
.000
|
a) State the linear regression equation.
b) Can you interpret the meaning of the y-intercept in the context of this study? Explain.
c) Interpret the slope of the regression equation in the context of these variables.
d) Perform the hypothesis test for the slope, and give your conclusion in words.
Question 8. An experiment with three different grips was performed to see what effect the grip might have on the distance of a backhanded Frisbee throw. The boxplots and ANOVA table for the three grips are shown below.
a) From examining the boxplots, do any of the grips appear to help throw the Frisbee significantly further than any other grip? Explain briefly.
b) State the hypotheses about the grips.
c) Give the values of the test statistic and P-value, and your conclusion for the test. Use α = 0.05.
d) Would it be appropriate to perform a Bonferroni test to see if any of the grips differ in the mean distance thrown?
Question 9. Chest deceleration measurements were made on dummies in car crash tests involving small, medium and large cars to investigate whether the size of the car was related to crash trauma. Smaller measurements correspond to less trauma to the crash test dummies. The data are given in the following table.
|
Size
|
Measurements
|
| Small
|
44
|
43
|
44
|
54
|
38
|
|
43
|
42
|
45
|
44
|
50
|
|
Medium
|
41
|
43
|
47
|
37
|
44
|
|
49
|
41
|
42
|
43
|
34
|
| Large
|
32
|
38
|
37
|
38
|
43
|
|
37
|
45
|
33
|
45
|
42
|
a) State the null and alternative hypotheses for the ANOVA test.
b) Use SPSS to carry out the ANOVA, and report the test statistic and p-value.
c) Write a brief conclusion of the ANOVA analysis.
d) Carry out the Bonferonni's multiple comparisons and summarise the results, if appropriate.