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SIN502S- STATISTICAL INFERENCE 1 - JAN 2020 |
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MAMIBIA UNIVERSITY
OF SCIENCE AND TECHMOLOGY
Faculty of Health and Applied Sciences
Department of Mathematics and Statistics
QUALIFICATIONS: BACHELOR OF SCIENCES IN APPLIED MATHEMATICS AND STATISTICS
QUALIFICATION CODE: 07BAMS
LEVEL: 5
COURSE: STATISTICAL INFERENCE 1
COURSE CODE: SIN502S
DATE: JANUARY 2020
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
SECOND OPPORTUNITY/SUPPLEMENTARY EXAMINATION QUESTION PAPER
EXAMINER(S)
MR. EM. MWAHI
MODERATOR:
DR. D. NTIRAMPEBA
THIS QUESTION PAPER CONSISTS OF 6 PAGES
(Including this front page)
INSTRUCTIONS
1. Answer all the questions and number your solutions correctly.
2. Question 1 of this question paper entails multiple choice questions with options A to
D. Write down the letter corresponding to the best option for each question.
3. For Question 2 & 3 you are required to show clearly all the steps used in the
calculations.
4. All written work MUST be done in blue or black ink.
5. Untidy/ illegible work will attract no marks.
PERMISSIBLE MATERIALS
1. Non-Programmable Calculator without the cover
ATTACHMENTS
Z-table, t-table, Chi-square table, Mann-Whitney U table and the F-table
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QUESTION 1 [20 MARKS]
1.1 Decreasing the confidence level, while holding the sample size the same, will
do what to the length of your confidence interval?
[2]
A. Make it bigger
B. Make it smaller
C. It will stay the same
D. Cannot be determined from the given information
1.2 If you increase the sample size, what will happen to the length of your
confidence interval?
[2]
A. Make it smaller
B. Make it bigger
C. It will stay the same
D. Cannot be determined from the given information
1.3 A certain brand of jelly beans are made so that each package contains about
the same number of beans. The filling procedure is not perfect, however. The
packages are filled with an average of 375 jelly beans, but the number going
into each bag is normally distributed with a standard deviation of 8. Yesterday,
Jane went to the store and purchased four of these packages in preparation for
a Spring party. Jane was curious, and she counted the number of jelly beans
in these packages - her four bags contained an average of 382 jelly beans.
1.3.1 In the above scenario, which of the following is a parameter?
[2]
A. The average number of jelly beans in Jane’s packages, which is 382.
B. The average number of jelly beans in Jane’s packages, which is unknown.
C. The average number of jelly beans in all packages made, which is 375.
D. The average number of jelly beans in all packages made is unknown
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1.3.2 If you went to the store and purchased six bags of this brand of jelly beans,
what is the probability that the average number of jelly beans in your bags is
less than 373?
[2]
A. 0.2709
B. 03085
C. 0.4013
D. 0.7291
1.4 A survey was conducted to get an estimate of the proportion of smokers among
the graduate students. Report says 38% of them are smokers. Chatterjee
doubts the result and thinks that the actual proportion is much less than this.
Choose the correct choice of null and alternative hypothesis Chatterjee wants
to test.
[2] ——p
A. Ho: p=0.38 versus Ha: p<0.38.
B. Ho: p=0.38 versus Ha: p >0 .38.
C. Ho: p=0.38 versus Ha: p< 0.38.
D. None of the above.
1.5 To test for equality of two population variances, one would use the__ test.
[2]
A. Zz
B. t
C. Chi-square
D. F
1.6 What test can be used to test the difference between two small sample means
when population variances are unknown?
[2]
A. Z
B. t
C. Chi-square
D. F
1.7 If in a random sample of 400 items, 88 are found to be defective. If the null
hypothesis is that 20% of the items in the population are defective, what is the
value of the test statistic?
[2]
A. 0.02
B. 1
C. 0.9656
D. 0.22
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1.8 A two-tailed test is one where:
[2]
A. Results in only one direction can lead to rejection of the null hypothesis
B. Negative sample means lead to rejection of the null hypothesis
C. Results in either of two directions can lead to rejection of the null
hypothesis
D. No results lead to the rejection of the null hypothesis
1.9 The null and alternative hypotheses divide all possibilities into:
[2]
A. Two sets that overlap
B. Two non-overlapping sets
C. Two sets that may or may not overlap
D. As many sets as necessary to cover all possibilities
QUESTION 2 [43 Marks]
2.1 MNM Corporation gives each of its employees an aptitude test. The scores on
the test are normally distributed with a mean of 75 and a standard deviation of
15. Asimple random sample of 25 is taken from a population of 500.
(a) What is the probability that the average aptitude test score in the sample
will be between 70.14 and 82.14?
[6]
(b) What is the probability that the average aptitude test score in the sample
will be equal to or greater than 82.68?
[4]
(c) Find a value, C, such that P(X > C) = 0.015.
[5]
2.2 A polling firm samples 600 likely voters and asks them whether they favour a
proposal involving school bonds. A total of 330 of these voters indicate that they
favour the proposal.
(a) Estimate the true population proportion of voters who favour the
proposal with a 99% level of confidence.
[5]
(b) | Can we conclude at the 10% level of significance that more than 50% of
all likely voters favour the proposal?
[5]
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2.3 In a study of the relationship of the shape of a tablet to its dissolution time, 6
disk-shaped ibuprofen tablets and 8 oval-shaped ibuprofen tablets were
dissolved in water. The two population variances are assumed to be nearly
equal. The dissolve times, in seconds, were as follows:
Disk: 269.0
249.3
255.2
252.7
247.0
261.6
Oval: 268.8
289.4
260.0
261.6
273.5
280.2
253.9
278.5
(a) _ Estimate and interpret the true population mean difference between the
two shapes of a tablet with the 5% level of significance.
(b) | Can we conclude that the mean dissolve times differ between the two
shapes? Use alpha = 0.01.
QUESTION 3 [26 Marks]
3.1 Does physical exercise alleviate depression? We find some depressed people
and check that they are all equivalently depressed to begin with. Then we
allocate each person randomly to one of two groups: 20 minutes of jogging per
day; or 60 minutes of jogging per day. At the end of a month, we ask each
participant to rate how depressed they now feel, on a Likert scale that runs from
1 ("totally miserable") through to 100 (ecstatically happy"). Ratings were
recorded in the table below:
Jogging for 20 minutes 22
27
39
29
46
48
49
Jogging for 60 minutes 59
66
38
49
56
60
56
(a) Use the Mann-Whitney U test to test if there is a difference in ratings
between the two groups. Use alpha = 0.01
(b) Suppose the data meet the requirements for a parametric test, what
parametric test can be used instead of the Mann-Whitney U test?
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32 The Mozart effect refers to a boost of average performance on tests for
elementary school students if the students listen to Mozart's chamber music for
a period of time immediately before the test. In order to attempt to test whether
the Mozart effect actually exists, an elementary school teacher conducted an
experiment by dividing her third-grade class of 6 students into three groups of
2. The first group was given an end-of-grade test without music; the second
group listened to Mozart's chamber music for 10 minutes; and the third groups
listened to Mozart’s chamber music for 20 minutes before the test. The scores
of the 15 students are given below:
Group1 Group 2 Group 3
80
79
73
63
73
82
Using the ANOVA F-test at a=0.10, is there sufficient evidence in the data to
suggest that the Mozart effect exists?
[12]
QUESTION 4 [11 MARKS]
In preparing a national promotional campaign to raise funds for Operation Feed
the Poor, the organising charity examined previous record of donations to
establish if age of donor is a factor in the monetary size of the donation received
from the donor. Their records were arranged into the following contingency
table:
Size
of | Age group
donation
20-34
Above $100 | 25
$50-$100
69
Under $50
36
35-49
40
51
29
50-64
47
74
19
Over 64
46
5/
37
Can it be concluded that the age of a donor influences the size of the donation
to this charity? Test at the 1% significance level.
[11]
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Critical Values of the Mann-Whitney U
(Two-Tailed Testing)
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APPENDIX C: The Standard Normal Distribution
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APPENDIX D: The t-distribution
‘ip
0.324920
0.288675 | 0.816497
0.276671 —_ 0.764892
40697
0.717558
0.263167 0.711142
1.533206
0.05
0.025
6.313752 —-12.70620-—«31.82052
(2.919986 4.30265 6.96456
2.353363 © «3.18245 4.54070
2.131847 2.77645 3.74695
2.015048
3.36493
(1.943180 2.
3.14267
+1,894579
"2.99795
«3.65674. 636.6192
315991
.
0.261921
0.706387
0.702722
1.859548 2.30600
2.89646
1.833113 2.26216 ~—- 2.82144
0.699812
(0.69744
(1.812461
1.795885
2.76377
2.71808
12
0.695483
- 1.782288
2.68100
13° (0.258591 _ 0.693829.
/1.770933
2.65031
14 (0.258213 -—:0.692417
1.761310
2.62449
0.257885 0.691197
1.753050 2.13145 «2.60248
0.257599 0.690132 1.336757 «1.745884 2.11991 «2.58349
17 0.257347 (0.689195 «1.333379 «1.739607 2.10982 «2.56693
18 0.257123 0.688364 (1.330391 1.734064 2.10092. 2.55238 += - 2.87844 «(3.9216
19 0.256923 0.687621. ~—«1.327728 —=«i1.729133
2.53948 2.86093 3.8834
20 0.256743 ~—<0.686954
1.724718
2.52798 ——- 2.84534 3.8495
21 0.256580 0.686352.
/1.720743
(2.83136 «3.8193
22 0.256432 ~—«0.685805.
1.717144
23. 0.256297 (0.685306 ~—«1.319460—1.713872
24 0.256173 0.684850 «1.317836 ~—=«1.710882_
2.49987
2.49216
25 0.256060 0.684430 «1.316345 «1.708141 2.05954 ~—-2.4851
26 0.255955 0.684043 «1.314972 «1.705618 2.05553 —«2.47863
27 0.255858 _ 0.683685 1.313703 1.703288 |
"2.47266
28 0.255768 ~—*0.683353. 1.312527 1.701131
2.46714
29 0.255684 —0.683044~—s«.311434
30 0.255605 0.682756
1.697261
2.46202
(2.45726
inf —_ 0.253347 0.674490
(1.644854 1.95996 2.32635 ~—=«.2.57583 «3.2905
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APPENDIX E: The Chi-Square Distribution
"0.00098
‘0.05064
(0.21580
0.48442
0.83121
500
0.10153 0.45494 1.32330
0.57536 |
| 13.27670
/-15.08627
“8.90652
/ 9.59078
23 9.26042
9.88623 10.
13.56471
14.25645
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2 Pages 11-20 |
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