Download sstt122 Test 2. Memorandum

TEST 2 – 24 OCTOBER 2013
FACULTY OF SCIENCE & AGRICULTURE
DEPARTMENT OF MATHEMATICAL SCIENCES
SSTT 122- Elementary Statistics for Commerce Students
DURATION: 3 HOURS
MARKS: 100
Lecturers
Miss J. M. Batidzirai
Mr. L. M. Sikhosana
INSTRUCTION TO CANDIDATES
1. Please ascertain that this question paper has Twenty (20) pages.
2. The paper consists of 40 Questions.
3. Answer all questions.
4. All questions should be answered on the provided answer sheet.
5. Calculators may be used.
6. Formulas are provided on APPENDIX A.
7. Statistical tables are provided on APPENDIX B.
8. Write neatly.
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1) The probability of the intersection of two independent events is
A. zero.
B. given by probability of their union.
C. given by product of their probabilities.
D. the probability of the complementary of their union.
E. None of the above
2) A researcher wants to determine whether the information medium people find most
trustworthy is dependent on the level of education. A sample of 400 people were
interviewed and their responses recorded in the following table:
EDUCATION
MEDIUM
Newspaper Television
Radio
Primary
education
45
22
6
High school
94
115
10
University
49
52
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Which of the following is INCORRECT?
A. The probability that a randomly selected person finds reading newspaper as trustworthy
is 0.47
B. The probability that a randomly selected person’s level of education is high school and
he finds reading newspaper as trustworthy is 0.235
C. The probability that a randomly selected person’s level of education is high school or he
finds reading newspaper as trustworthy is 0.7825.
D. The probability that a randomly selected person finds reading newspaper
trustworthy and he finds watching television trustworthy is 0.47
E. None of the above
3) From the information provided in Question 2 above, what is the probability that a randomly
selected person’s level of education is high school given that he finds reading newspaper as
trustworthy
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A. 0.5
B. 0.5956
C. 0.47
D. 0.2875
E. None of the above
4) Which of the following is CORRECT?
A. For two independent events, the occurrence of one event has no effect on the occurrence
of the other.
B. Two mutually exclusive events have nothing in common.
C. The sum of all probabilities in the sample space is 1.
D. All of the above.
E. None of the above.
5) Which of the following is INCORRECT?
A. For a Poisson distribution, 𝐸(𝑋) = 𝑉𝑎𝑟(𝑋)
B. For a binomial distribution, 𝑉𝑎𝑟(𝑋) = 𝑛𝑝(1 − 𝑝)
C. The probability mass function for a Poisson distribution is P(Y  y) 
e   y
.
y!
D. All of the above.
E. None of the above.
6) If the random variable, 𝑋 , has a binomial distribution with 7 observations, each with a
probability of success as 0.3 then 𝑃(𝑋 > 1) =
A. 0.9176
B. 0.0824
C. 0.3294
D. 0.6706
E. None of the above
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7) The probability distribution of 𝑋 , where 𝑋 is the average number of cheques received at a
bank per day, is as follows:
𝒙
𝑷(𝑿 = 𝒙)
0
1
2
3
0.35
0.35
0.05
0.25
The following calculations were made:
i. The expected number of bad cheques at a bank per day is 1.2
ii. 𝑋~𝐵𝑖𝑛(3)
iii. 𝑃(𝑋 > 2) = 0.3
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iv.
 P( X  x)  1
x 0
Which of the statements above are true?
A. i
B. i and iv
C. i, ii and iv
D. All of the above
E. None of the above
8) From Question 7 above, what is the value of the standard deviation?
A. 2.8
B. 1.6
C. 1.36
D. 1.16619
E. None of the above
9) Cumulative distribution function is defined as 𝑃(𝑋 ≤ 𝑥) =
A. ∑𝑘<𝑥(𝑋 = 𝑘)
B. ∑𝑥∈𝑅 𝑃(𝑥)
C. ∑𝒌≤𝒙 𝑷(𝑿 = 𝒌)
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D. ∑ 𝑥𝑃(𝑥)
E. None of the above
10) Which of the following situations does follow a binomial distribution?
A. Number of car accident deaths recorded per month.
B. The average breakdowns per day of fleet trucks.
C. Five independent contracts each with a probability of making a sale of 0.4.
D. The mean number of strikes per year.
E. None of the above
11) A certain scientific theory supposes that mistakes in cell division occur according to a
Poisson process with rate 2.5 per year, and that an individual dies when 196 such mistakes
have occurred. Assuming this theory find the mean lifetime of an individual
A. 490 years.
B. 2.5 years.
C. 78.4 years.
D. Cannot be calculated from the given information.
E. None of the above
12) Which of the following is INCORRECT?
A. In a normal distribution, the 𝑷(𝒁 < 𝟏. 𝟖) = 𝑷(𝒁 > 𝟏. 𝟖).
B. In a normal distribution, the 𝑃(𝑍 < −1.8) = 𝑃(𝑍 > 1.8).
C. In a normal distribution, the 𝑃(𝑍 < 1.8) = 1 − 𝑃(𝑍 > 1.8).
D. In a normal distribution, the 𝑃(𝑍 < 1.8) = 𝑃(𝑍 ≤ 1.8).
E. None of the above.
13) Which of the following statements is INCORRECT?
A. In a normal distribution the histogram of the data is symmetric about the
variance.
B. In a normal distribution the histogram of the data is bell shaped.
C. In a normal distribution, the histogram of the data has one peak.
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D. All of the above are INCORRECT.
E. None of the above is INCORRECT
14) For a standard normal distribution,
A. 𝑋~𝑁(𝜇; 𝜎 2 )
B. 𝑋~𝑁(𝜇; 𝜎)
C. 𝑍~𝑁(0; 1)
D. 𝑍~𝑁(𝜇; 𝜎 2 )
E. None of the above is CORRECT.
15) Which of the following is INCORRECT about the central limit theorem
A. Applies to large samples, 𝑛 > 30.
 2 
B. For large samples, X ~ N  ; 
n 

C. For large samples, X ~ N   ;  2 
D. For large samples, Z 
X 
2
~ N (0;1) .
n
E. None of the above
16) Find the value of 𝑘, given that 𝑃(𝑘 < 𝑍 < 1.8) = 0.78
A. 0.9
B. -0.9
C. 1.96
D. -1.96
E. None of the above
17) Which of the following statements is INCORRECT?
A. Type I error is committed by rejecting a correct null hypothesis.
B. Probability of committing a type I error is the level of significance.
C. The critical value divides the non-rejection region from the rejection region.
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D. Type II error is committed by rejecting an incorrect null hypothesis.
E. None of the above.
18) Which of the following is INCORRECT concerning hypothesis testing?
A. The null hypothesis is rejected if calculated value of the test statistic greater than the
critical value
B. The significance level is the maximum probability that we will make a Type I error.
C. The t-test statistic is used where a sample mean is involved, sample size is small and the
population variance is unknown.
D. The z-test statistic is used where independence of two variables is tested.
E. None of the above.
19) Degrees of freedom for a Chi- square distribution are given by
A. Sample size -1.
B. Sample size – 2.
C. (Number of Rows-1) * (Number of columns-1).
D. (Number of Rows) * (Number of columns).
E. None of the above.
20) Which of the following does NOT apply about the confidence interval of the population
mean, where population standard deviation is unknown and the sample is small?
𝜎2
A. 𝜇 = 𝑥̅ ± t  n  1 √ 𝑛
2
𝝈𝟐
̅ ± 𝒁𝜶 √
B. 𝝁 = 𝒙
𝒏
𝟐
𝜎2
C. 𝜇 = (𝑥̅ − t  n  1 √ 𝑛 ; ∞)
𝜎2
D. 𝜇 = (−∞; 𝑥̅ + t  n  1 √ 𝑛 )
E. None of the above
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21) Given the following graph for a standard normal distribution:
Which of the following probabilities is represented by the shaded area?
A. 𝑃(𝑍 ≤ 𝑎) = 0.05
B. 𝑷(𝒁 ≥ 𝒂) = 𝟎. 𝟎𝟓
C. 𝑃(−𝑎 ≤ 𝑍 ≤ 𝑎) = 0.05
D. 𝑃(𝑍 > 𝑎) = 0.95
E. None of the above
22) In the graph given in Question 21 above, find the value of a?
A. ±1.96
B. ±1.645
C. 1.96
D. 1.645
E. None of the above
23) For a normal distribution, P(a  Z  b) 
A. P( Z  b)  P( Z  a )  1
B. P( Z  b)  P( Z  a)
C. P( Z  a)  P( Z  b)
D. P( Z  b)  P( Z  a)
E. All of the above are true
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24) Let the masses of airline passengers have a distribution with mean 80kg and a standard
deviation of 30kg. A certain light aircraft company sets a 4500kg limit for the total mass of
the passengers on board. What is the probability that a flight of 50 passengers will overload
it?
A. 0.0091
B. 0.3707
C. 0.9909
D. Cannot be calculated from the given information.
E. None of the above.
25) Which distribution is suitable for the test of independence of 2 qualitative variables?
A. t- distribution.
B. Normal distribution.
C. Chi- square distribution.
D. F- distribution.
E. None of the above.
USE THE GIVEN INFORMATION FROM THE TABLE IN QUESTION 2 ABOVE,
TO PERFORM A HYPOTHESIS TESTING IN QUESTIONS 26-31.
26) Which of the following is a proper null hypothesis?
A. The information medium people find most trustworthy is dependent on the level of
education
B. The information medium people find most trustworthy is independent on the level
of education
C. The information medium people find most trustworthy is mutually exclusive to the level
of education
D. The information medium people find most is not mutually exclusive to education
E. None of the above
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27) Suppose we are testing the hypothesis at 0.01 level of significance, what is the critical value
for this test?
A. 112.3
B. 13.28
C. 7.779
D. 21.697
E. None of the above.
28) To calculate the test statistic, contingency table was formed and the following expected
values were calculated for each cell:
EDUCATION
MEDIUM
Newspaper
Television
Radio
Primary
X
34.4925
4.1975
High school
102.93
Y
12.5925
University
50.76
51.03
Z
education
Which of the following values are correct for filling in the table?
A. X = 54.4, Y = 115.478 and Z = 6.21
B. X = 34.31, Y = 123.478 and Z = 6.21
C. X = 54.4, Y = 115.478 and Z = 6
D. X = 34.31, Y = 103.478 and Z = 6.21
E. None of the above
29) Which of the following is the correct test statistic?
A. 11.40
B. 23.21
C. 8.3816
D. 191.40
E. None of the above
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30) Suppose the 4 Questions above are all A, what will be the correct decision to make?
A. Since the observed value is greater than the critical value we reject 𝐻𝑜
B. Since the observed value is greater than the critical value we do not reject 𝐻𝑜
C. Since the observed value is less than the critical value we reject 𝐻𝑜
D. Since the observed value is less than the critical value we do not reject 𝑯𝒐
E. None of the above.
31) ) Suppose the solutions to the above 5 Questions are all D, what will be the CORRECT
conclusion?
A. The information medium people find most trustworthy is dependent on the level of
education.
B. The information medium people find most trustworthy is mutually exclusive to the level
of education.
C. The information medium people find most is not mutually exclusive to education.
D. The information medium people find most trustworthy is independent on the level
of education.
E. None of the above
32) Given the following statements:
i.
To find tcritic , we must know the degrees of freedom as well as the level of
significance.
ii.
Cumulative Distribution function is the sum of all the probabilities from the smallest
observation to a specified observation of x .
iii.
E  X    x2 P  X  x 
x
Which of the 3 statements above is/ are CORRECT?
A. i and iii
B. ii and iii
C. i and ii
D. All of the above
E. None of the above
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USE THE FOLLOWING INFORMATION TO ANSWER QUESTIONS 33- 38.
The performance of 10 salesmen (measured in thousands of Rands of sales) is evaluated
before and after an intensive training on selling techniques. A hypothesis testing was done
at 10% level of significance to test whether there has been a significant increase in the
number of sales after the intensive training. The following results were obtained:
Performance
Salesman before the course
(in thousands of
Rands)
1
10.4
2
22.3
3
24.1
4
42.1
5
37.5
6
77.9
7
99.5
8
96.3
9
89.5
10
85.4
Performance after
the course
(in thousands of
Rands)
12.3
20.3
21.7
35.6
32.8
60.5
62.3
55.3
40.8
38.3
33) Formulate the correct hypotheses:
A. 𝐻0 :𝜇𝐷 > 0 (There has been a significant increase in the sales after the intensive
training)
𝐻1 : 𝜇𝐷 > 0 (There has been no significant increase in the sales after the intensive
training)
B. 𝐻0 :𝜇𝐷 = 0(There has been no significant increase in the sales after the intensive
training)
𝐻1 : 𝜇𝐷 > 0 (There has been a significant increase in the sales after the intensive
training)
C. 𝑯𝟎 : 𝝁𝑫 = 𝟎 (There has been no significant increase in the sales after the intensive
training)
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𝑯𝟏 : 𝝁𝑫 < 𝟎 (There has been a significant increase in the sales after the intensive
training)
D. 𝐻0 : 𝜇𝐷 = 0 (There has been a significant increase in the sales after the intensive
training)
𝐻1 : 𝜇𝐷 < 0 (There has been no significant increase in the sales after the intensive
training)
E. None of the above is the correct hypothesis
34) Which distribution is appropriate for testing this hypothesis?
A. Normal distribution
B. Chi- Square distribution
C. t- distribution
D. Fisher’s distribution
E. None of the above
35) What is the correct absolute value of the test statistic?
A. 5.368
B. -3.1445
C. 6.9667
D. 3.1445
E. None of the above
36) What is the correct absolute value of the critical value?
A. 1.183
B. 1.383
C. 1.833
D. 1.372
E. None of the above
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37) Which of the following decisions is correct, given that the answers to question 34, 35 and
36 are all A.
A. Reject 𝑯𝟎
B. Do not reject 𝐻0
C. No decision
D. All of the above
E. None of the above
38) Which of the following conclusion is correct, given that the answers to question 34, 35, 36
and 37 are all A.
A. There has been a significant increase in the sales after the intensive training
B. There has been no significant increase in the sales after the intensive training
C. All of the above are correct.
D. None of the above.
39) An estimated range which is likely to include an unknown population parameter
A. Interquartile range
B. Range.
C. Error range.
D. Confidence interval.
E. None of the above.
40) Given that 𝑃(𝐴) = 0.3 and 𝑃(𝐵′ /𝐴′ ) = 0.4. Find 𝑃(𝐴 ∪ 𝐵).
A. 0.12
B. 0.28
C. 0.72
D. Cannot be found by the insufficient information given.
E. None of the above
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