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BEE1022

INTRODUCTION TO STATISTICS

2022

Section A    [45 marks]

Answer all questions. There is exactly one correct answer to each question. Each question carries 2.5 marks.

1.  A fair coin is flipped 3 times. The probability of 2 heads occurring is approximately

A.  0.333

B.  0.667

C. 0.375

D. 0.500

E.  None of the above

2.  Given a standard normal distribution, the area under the curve between z = 0 and z = 2.50 is equal to

A.  2.500

B.  0.006

C. 0.994

D. 0.494

E.  None of the above

3.  The mean daily studying time of all university students is 120 minutes, with a standard     deviation of 10 minutes. What is the probability that a randomly chosen university student will be studying less than 110 minutes in a day?

A.  0.1587

B.  0.8413

C. 0.9990

D. 0.001

E.  None of the above

4.  Consider the following data on unemployment rate (% share of total workforce) in selected regions in the year 2020.

Region

Unemployment

LCN

10.53

TLA

10.53

LAC

10.51

NAC

8.21

ECA

7.87

EMU

7.84

ECS

7.42

TEC

7.38

OED

7.25

EUU

7.05

PST

6.9

HIC

6.68

EAR

5.98

SAS

4.84

TSA

4.84

What is the 75th percentile of unemployment rate in this sample?

A.  8.04

B.  6.67

C. 6.9

D. 6.68

E.  None of the above

For questions 5 to 7, consider the following 2 x 4 contingency table in which a population of 3129 companies is summarised in terms of the companies’ size (micro, small, medium, and large) and type of ownership (domestic and foreign).

 

Micro

Small

Medium

Large

Domestic

423

576

400

602

Foreign

102

102

357

567

5.  What is the probability that a randomly drawn company is small or medium, given that is foreign owned?

A.  0.0904

B.  0.3165

C. 0.4069

D. 0.1467

E.  None of the above

6.  What is the probability that a randomly drawn company is of medium size or domestically owned?

A.  0.1278

B.  0.2464

C. 0.7536

D. 0.8814

E.  None of the above

7.   Which of the following statements is correct?

A.  Company size and ownership are independent

B.  Company size and ownership are mutually exclusive

C. Company size and ownership are not independent

D. There is not enough information to decide whether company size and ownership are independent or not

E.  None of the above are correct.

8.   Suppose it is known that on average, 4 people per minute call a company’s customer services hotline. What is the probability that fewer than two calls are received in one   minute?

A.  0.1465

B.  0.2381

C. 0.0916

D. 0.1954

E.  None of the above

9.   How many different combinations of a 5-member basketball team can be formed from a group of 10 qualified players?

A.  50

B.  252

C. 522

D. 2

E.  Not enough information to decide

10. Suppose 30% of all countries prefer to have emission trading as an instrument to bring     down their carbon emission, while 50% of them prefer to have carbon tax. Suppose further

that 20% of countries prefer to have both instruments. If we randomly choose a country, what is the probability that the country would prefer to have neither emission trading nor carbon tax?

A.  0.2

B.  0.4

C. 0.6

D. 0.8

E.  1

11. Let X be a Normally distributed random variable with mean 1 and standard deviation 1. The 60th percentile of X  is _____.

A.  equal to 1

B.  equal to 0

C. equal to 0.24

D.  in between 1.25 and 1.26

E.  in between 0.25 and 0.26

12.  Let X be normally distributed with mean 25 and variance 9. What is the (approximate) inter-quartile range (IQR) of this distribution?  (2 decimal points)

[Hint: use the table for the standard Normal distribution at the end of the exam paper. Take the values closest to the ones you are looking for from the table. Do not       interpolate. Hence your IQR will be an approximation.]

A.  4.02

B.  54.02

C. 45.98

D. 6.02

E.  None of the above

13.  A random sample of n = 64 is obtained from a population with a known variance  2  =

576 and unknown mean u . The sample average is found to be 250. What can you say about the distribution of  ?

A.   is t distributed with 63 degrees of freedom.

B.   is exactly normally distributed with mean 250 and variance 9.

C.  is approximately normally distributed with mean 250 and variance 9.

D.  is approximately normally distributed with mean 250 and variance 3.

E.  None of the above.

14.  Suppose that the 95% confidence interval for the mean has been estimated as [−4.6 ,  5.3] . Which of the following statements is true?

A.  The sample mean lies inside this interval with 95% probability .

B.  There is a 5% probability that the population mean lies outside this interval.

C. We can reject the null hypothesis that the mean is equal to zero at the 5% level of significance.

D. The confidence interval would be wider if the sample size was increased, holding everything else fixed

E.  None of the above are correct.

15.  Suppose we want to test the null hypothesis that the population mean hourly wage is     £8.50 against the alternative hypothesis that the population mean is less than £8.50. For the randomly selected 15 workers, we calculated that the average hourly wage is £8.40,  with a sample standard deviation is £3.2. Let us assume that the population hourly wage  is normally distributed. The level of significance is selected as a = 0.05. The decision rule for this test is to reject the null hypothesis if the test statistic is ________.

A.  less than -1.761 or greater than 1.761

B.  less than -2.145 or greater than 2.145

C.  less than -1.753

D.  less than -1.761

E.  greater than -1.761

16.A researcher wants to know whether firm-size (micro, small, medium, large) is dependent on the ownership (foreign-owned or domestic-owned). The appropriate test and the critical value at a = .01 are ________ and ______, respectively.

A.  Goodness-of-fit test and 11.345

B.  Test for independence and 11.345

C. Goodness-of-fit test and 15.086

D. Test for independence and 15.086

E.  None of the above

17.Suppose we have two independent random variables: X1 ~N(5, 10) and X2 ~N(3,8). What is the distribution of Y =  2X1  + 3X2 ?

A. Y N(8, 18)

B. Y N(8, 44)

C. Y N(19, 112)

D. Y N(19, 44)

E. Y N(19, 164)

18. Consider the following table, which shows two random samples of grades obtained from two groups of students.

Group A

Group B

70

60

65

72

43

57

50

48

85

91

90

46

42

82

 

43

Assume that grades are normally distributed in the population, and that SA(2)  > SB(2) .

You want to conduct a hypothesis test where the null hypothesis is H0 : GA2  = GB(2)  and the alternative hypothesis is  H0 : GA2  ≠ GB(2)  at the 5% level of significance. What is the upper critical value for this test?

A.  3.87

B.  5.12

C. 5.7

D. 4.21

E.  4.53

Section B   [55 marks]

Answer all questions.

Question 1   [15 marks]

You are given data on labour productivity for a sample of 28 countries:

Labour

102

106

120

134

productivity:

118

 

101

 

107

 

115

108

 

96.8

 

96.2

 

116

107

 

108

 

103

 

98.3

108

 

113

 

111

 

106

113

 

139

 

115

 

107

126

 

118

 

98.8

 

102

Answer the following questions. Show all your workings.

a)  [6 marks]  Find the mean (2dp), median and 29th percentile of labour productivity in this sample. Describe in your own words what the 29th percentile is.

b)  [3 marks]  Find the interquartile range.

c)  [6 marks]  Sketch a box-and-whiskers plot. Label all elements carefully.

Question 2   [15 marks]

Ari and Lisa are keen tennis players. They play each other once a week on an outdoor tennis    court. When it rains, they have to use an indoor court instead. Lisa wins 2 out of 3 matches they play against each other indoors, and Ari wins 6 out of 10 matches they play outdoors.                The weather forecast for the afternoon gives a 30% chance of rain.

a)  [6 marks] Draw a tree diagram for this problem.

b)  [3 marks] What is the probability that Ari wins today’s match? (2 decimal places)

c)  [3 marks] Find Pr(match is indoors | Lisa wins)  (3 decimal places)

d)  [3 marks] Find Pr(match is indoors U Lisa wins)  (2 decimal places)


Question 3   [25 marks]

A researcher wants to test if the way in which food is served has an impact on the perceived  quality of the food. In her study, food is served either on regular china, on single-use plastic   plates, or on re-usable plastic plates. Which plate is used is decided randomly for each order. After the end of their meal, all diners are asked if they would recommend their food to a friend (yes or no).

Her results are as follows:

 

regular

china             single-use plastic

re-usable

plastic

 

total

 

recommend

 

 

do not recommend

 

63

 

 

14

 

45

 

 

24

 

62

 

 

31

 

170

 

 

69

a)  [4 marks]  In your opinion, does this study constitute a randomised control trial? Explain.

b)  [14 marks]  Conduct the appropriate hypothesis test to answer the research question. Use the 5% level of significance. Follow all the steps carefully and show all your workings.       Round all results to 2 decimal places.

 

c)  [7 marks]  Based on the total number of diners in this sample, find the 95% confidence      interval for the population proportion of diners who would recommend their food to a friend (3 decimal places).