辅导 ECMT1010 Introduction to Economic Statistics End of Semester Examination Semester 1 – 2016辅导 C/C+

ECMT1010

Introduction to Economic Statistics

End of Semester Examination

Semester 1 – 2016

30 Multiple Choice Questions [ 15 marks total—suggested time approx. 32 minutes].

1. A bank reports that 30% of households have a MasterCard, 20% have an American Express card, and 25% have a Visa card. Eight percent of households have both a MasterCard and an American Express card. Twelve percent have both a Visa card and a MasterCard. Six percent have both a American Express card and a Visa card. If a household has a MasterCard, what is the probability it also has a Visa card?

A) 0.12

B) 0.25

C) 0.40

D) 0.43

E) 0.48

2. Lenovo Group Limited, a Hong Kong IT company, has a 30% share of the Hong Kong PC market. Suppose 10 new PC buyers are selected at random from the Hong Kong population. What is the probability that fewer than 3 bought their PC from Lenovo?

A) 0.028

B) 0.121

C) 0.233

D) 0.267

E) 0.382

3. A pair of (fair) dice is rolled once. What is the probability that the sum of the values on the two die faces is 7?

A) 1/6

B) 7/36

C) 1/2

D) 1/18

E) 1/3

4. Calculate the expected value (mean) of the following discrete probability distribution:

x 1 2 3 4

p(x ) 0.16 0.26 0.26 0.32

A) 1

B) 2.5

C) 1.86

D) 2.74

E) 2.0

5. A psychologist is interested to test whether the IQ of statisticians is higher than 100. Based on a random sample of 100 statisticians, the sample mean of IQ is 120. What is the p-value of the test assuming a population standard deviation of 100?

A) 0.0000

B) 0.0228

C) 0.0456

D) 0.4207

E) 0.8414

Scenario 1 In a marketing research project, a major supermarket wants to study the relationship between the annual consumption of ramen noodles (Y , in number of packs) and the annual income level of con- sumers (X , in $000s). Based on a random sample of 100 customers, the linear regression model

Yi = β0+ β1Xi + εi

is estimated with the following result:

Coefficient Standard Error

Intercept 55.4 32.3

Annual income (in $000s) −0.22 0.1

6. Refer to Scenario 1. What is the predicted annual consumption (in number of packs) of ramen noodles for a consumer who earns $100,000 a year?

A) 2.2

B) 22.0

C) 33.4

D) 53.2

E) 55.4

7. Refer to Scenario 1. To see whether income level has an effect on the consumption of ramen noodles, Adam, Simon and Tim consider the following hypotheses:

H0 : β1 = 0 Ha : β1 0

They arrive at the following conclusions:

Adam: The null hypothesis is rejected at the 5% signiﬁcance level. Simon: The null hypothesis is rejected at the 2% signiﬁcance level.

Tim: The null hypothesis is rejected at the 1% signiﬁcance level. Who is/are correct?

A) Tim only

B) Simon only

C) Adam only

D) Adam and Simon only

E) Adam, Simon and Tim

8. Alcohol content in beer is believed to follow a normal distribution. A chemist takes a sample from 9 bottles of beer and measures the alcohol content, ﬁnding a sample mean of 7.5% and a sample standard deviation of 1%. The chemist wishes to compute a 90% conﬁdence interval for the mean. However, the chemist mistakenly treats the sample standard deviation as if it were the population standard deviation. What is the conﬁdence interval constructed by the chemist?

A) (6.647, 8.153)

B) (6.952, 8.048)

C) (6.880, 8.120)

D) (7.073, 7.927)

E) (7.034, 7.966)

9. Suppose the chemist in the previous question realises he has made a mistake. If he correct his mistake andrecalculates the conﬁdence interval using the same sample, how will the new conﬁdence interval compare to the previous one?

A) The new interval will be the same width as the previous one and will be shifted to the left to account for small sample bias.

B) The new interval will be wider than the previous one and will be centered around the same point estimate.

C) The new interval will be narrower than the previous one and will be centered around the same point estimate.

D) The new interval will be narrower than the previous one and will be shifted to the left to account for small sample bias.

E) This cannot be determined from the data given.

10. A lecturer hires a tutor to mark exam papers. To ensure that the tutor is grading correctly, the lecturer marks a few exam papers herself and compares her mark with the mark given by the tutor. She chooses these papers by physically going through the pile of exams and pulling out a paper “when she feels like it” . This corresponds to which form. of sampling?

A) Judgement sampling

B) Simple random sampling

C) Systematic sampling

D) Cluster sampling

E) Snowball sampling

11. A pair of (fair) die is rolled once. What is the probability that the sum of the values on the two die faces is not a 7?

A) 1/6

B) 7/36

C) 1/3

D) 1/2

E) 5/6

Scenario 2 The following table is derived from the Banerjee et al (2010) study on vaccination rates in India.

Control Group Treatment Only Treatment plus Incentive

Children not fully immunised 810 311 234

Children fully immunised 50 68 148

12. Refer to Scenario 2. What is the point estimate of the proportion of children that were fully immu- nisedin villages that received only the treatment and not the additional incentives?

A) 0.613

B) 0.219

C) 0.179

D) 0.821

E) 0.387

13. Refer to Scenario 2. Suppose a researcher wants to test the null hypothesis that the true proportion of children that were fully immunised in villages that received only the treatment was exactly 0.2 at a 99% level of signiﬁcance. What critical value will the researcher have to look up in the appropriate statistical table in order to do this?

A) 1.28

B) 1.645

C) 1.96

D) 2.33

E) 2.575

14. Complete the following sentence to arrive at the correct statement of the Central Limit Theorem: “If samples of size n are drawn randomly from a population with mean μ and standard deviation σ . . . ”

A) then repeated observations of the sample mean will follow a normal distribution with mean μ and standard deviation σ, regardless of the underlying distribution.

B) then if the sample size is sufﬁciently large (n ≥ 30), repeated observations of the sample mean will follow a normal distribution with mean μ and standard deviation σ, regardless of the underlying distribution.

C) then if the sample size is sufﬁciently large (n ≥ 10), repeated observations of the sample mean will follow a normal distribution with mean μ and standard deviation σ/ √n, regardless of the underlying distribution.

D) then if the sample size is sufﬁciently large (n ≥ 30), repeated observations of the sample mean will follow a normal distribution with mean μ and standard deviation σ/ √n, regardless of the underlying distribution.

E) then if the sample size is sufﬁciently large (n ≥ 30), repeated observations of the sample mean will follow a normal distribution with mean μ and standard deviation σ/√n, as long as the underlying distribution is normal.

15. A researcher is interested in the following hypotheses about the mean of a population: H0 : μ ≤ 2 Ha : μ > 2

Based on a sample of 45 observations and the researcher calculates a t statistic of 2.2. At a 1% level of signiﬁcance what is the researcher’sconclusion?

A) The researcher is unable to reject a false null hypothesis.

B) The researcher fails to reject the null hypothesis.

C) The researcher accepts the null hypothesis.

D) The researcher rejects the null hypothesis.

E) The researcher commits a type I error.

Scenario 3 On the ﬁrst day of class, students in an introductory economics course were asked their sex and eye color. The results are summarized in the table below.

	Blue	Brown	Green	Hazel	All
Female	24	21	10	11	66
Male	20	17	8	10	55
Total	44	38	18	21	121

16. Refer to Scenario 3. What is the probability that a randomly selected student in the class is a female or has brown eyes?

A) 0.660

B) 0.860

C) 0.314

D) 0.545

E) 0.686

17. Refer to Scenario 3. What is the probability that a randomly selected student in the class is a female and has hazel eyes?

A) 0.634

B) 0.091

C) 0.149

D) 0.174

E) 0.545

18. Refer to Scenario 3. What is the probability that a randomly selected student is a male, if we know that they have hazel eyes?

A) 0.476

B) 0.182

C) 0.083

D) 0.078

E) 0.455

19. An article published in the Canadian Journal of Zoology presented a method for estimating the body fat percentage of North American porcupines. The method was illustrated with a sample of n = 25 porcupines. Based on this sample, a 95% bootstrap conﬁdence interval for the average body fat percentage of porcupines is 17.4% to 25.8%. Which of the following null hypotheses would be rejected based on this conﬁdence interval?

A) H0 : μ = 18.6%.

B) H0 : μ = 26.6%.

C) H0 : μ = 20.0%.

D) H0 : μ = 22.9%.

E) H0 : μ = 24.6%.

Scenario 4 Admissions records at MIT indicates that 6.7% of the graduate students enrolled are from Canada.

20. Refer to Scenario 4. What is the minimum sample size for which the Central Limit Theorem applies in this case?

A) n = 30.

B) n = 40.

C) n = 50.

D) n = 100.

E) n = 200.

21. Refer to Scenario 4. Find the mean and standard error of the sample proportion of Canadian students in random samples of 100 graduate students at MIT.

A) p(ˆ) = 0.067, SE = 0.0625.

B) p(ˆ) = 0.067, SE = 0.006.

C) p(ˆ) = 0.067, SE = 0.025.

D) p(ˆ) = 0.670, SE = 0.250.

E) p(ˆ) = 0.067, SE = 0.0067.

22. Refer to Scenario 4. Roughly what percentage of samples of 100 randomly selected graduate students at MIT will have at least 10% of students from Canada?

A) 5%.

B) 6.7%.

C) 10%.

D) 18%.

E) 25%.

23. For a N(0, 1) density, what is the area to the left of z = -1.645.

A) 2.5%.

B) 3.5%.

C) 5%.

D) 10%.

E) 11%.

24. For a N(0, 1) density, what is the area outside of the interval z = -2.326 and z = 1.282.

A) 2.5%.

B) 3.5%.

C) 5%.

D) 10%.

E) 11%.

25. A sample of 148 university students reports sleeping an average of 6.85 hours on weeknights. The sample size is large enough to use the normal distribution, and a bootstrap distribution shows that the standard error is SE = 0.175. Use a normal distribution to construct a 95% conﬁdence interval for the mean amount of weeknight sleep students get at this university.

A) 6.68 to 7.03 hours.

B) 6.51 to 7.19 hours.

C) 6.50 to 7.20 hours.

D) 6.52 to 7.21 hours.

E) 6.85 to 7.85 hours.

26. Suppose that a 95% conﬁdence interval for μ is (54.8, 60.8). Which of the following is most likely the p-value for the test of H0 : μ = 56 versus Ha : μ 56?

A) 0.031

B) 0.001

C) 0.016

D) 0.231

E) 0.05

27. The randomization distribution for testing the hypotheses H0 : μ1 = μ2 versus Ha : μ1 μ2 is provided. The sample statistic is 1 - 2 = -2.5. Use the provided randomization distribution (based on 100 samples) to estimate the p-value for this test.

A) 10% B) 2% C) 5% D) 1% E) 4%

Scenario 5 Refer to the following probability tree diagram to ﬁnd the requested probabilities. (Round your answers to two decimal places.)

28. Refer to Scenario 5. What is P(YjA)?

A) 0.60

B) 0.50

C) 0.40

D) 0.20

E) 0.06

29. Refer to Scenario 5. What is P(AjY)?

A) 0.82

B) 0.30

C) 0.20

D) 0.18

E) 0.06

30. Refer to Scenario 5. What is P(X )?

A) 0.66

B) 0.50

C) 0.48

D) 0.42

E) 0.44

Problem 1 [20 marks total—suggested time approx. 44 minutes]

Using data from the United States for 1970–2009, a researcher obtains the following regression output for a model to predict life expectancy based on the total number of vehicles produced (measured in thousands).

Predictor Coefficient	SE coef .	t stat
Intercept 65 .8455	0 .2434	270 .5326
Vehicles 0 .05015	0 .001286	38 .9868
Regression statistics
R square 0 .9756	SD error	0 .3311 Observations 40
Analysis of variance
Source df	SS
Regression 1	166 .6377
Residual 38	4 .1660
Total 39	170 .8037