UNIVERSITY OF VICTORIA

Midterm October 19, 2017

solutions

NAME: _____________________________

STUDENT NUMBER: V00______________

Course Name & No. Statistical Inference

Economics 246

Section(s) A01

CRN: 10991

Instructor: Betty Johnson

Duration: 60 minutes This exam has a total of __ pages including this cover page.

Students must count the number of pages and report any discrepancy immediately to the

Invigilator.

This exam is to be answered: In Booklets provided

Marking Scheme: Part I:

Q1: 20 marks

Part II:

Q2: 10 marks

Part III:

Q3: 3 marks

Part IV:

Q4: 6 marks

Q5: 11 marks

Materials allowed: Non-programmable calculator

Econ 246 Summer 2017 CRN#: 10991

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Part I: Multiple choice and True/False 1. In a recent survey of college professors, it was found that the average amount of money

spent on food each week was normally distributed with a mean of $55.25 and a standard

deviation of $14.54. What is the probability that the average spending of a sample of 25

randomly-selected professors will exceed $59?

A) 0.0985

B) 0.9015

C) 0.0865

D) 0.0910

ANSWER: A

2. If a sample of size 64 is taken from a population whose standard deviation is equal to 48,

then the standard error of the mean is equal to

A) 6

B) 0.75

C) 288

D) None of the above

ANSWER: A

3. The time it takes to complete the assembly of an electric toaster is normally distributed

with a standard deviation of 4 minutes. If we randomly select 21 components, what is the

probability that the standard deviation for the time of assembly of these units is less than

4.0 minutes?

A) <0.010

B) <0.050

C) <0.025

D) <0.100

ANSWER: D

4. What is the name of the parameter that determines the shape of the chi-square

distribution?

A) The mean

B) The variance

C) The sample standard deviation

D) The degrees of freedom

ANSWER: D

5. Random samples of size 81 are taken from a population whose mean is 85 and standard

deviation is 27. The mean and standard error of the sampling distribution of sample

means are, respectively,

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A) 85 and 27

B) 85 and 3

C) 85 and 0.333

D) 9.444 and 27

ANSWER: B

6. The standard deviation of the sampling distribution of the sample mean is also called the

A) central limit theorem

B) standard error of the mean

C) finite population correction factor

D) population standard deviation

ANSWER: B

7. If all possible samples of size n are drawn from an infinite population with a mean of 35

and a standard deviation of 7, then the standard error of the sampling distribution of

sample means is equal to 1.0 only for samples of size

A) 7.

B) 14.

C) 49.

D) 2401.

ANSWER: C

8. If a random sample of size n is drawn from a normal population, then the sampling

distribution of sample means will be:

A) normal for all values of n

B) normal only for n > 30

C) approximately normal for all values of n

D) approximately normal only for n > 30

ANSWER: A

9. Why is the Central Limit Theorem important in statistics?

A) Because for a large sample size n, it says the population is approximately normal.

B) Because for any population, it says the sampling distribution of the sample mean is

approximately normal, regardless of the shape of the population.

C) Because for a large sample size n, it says the sampling distribution of the sample

mean is approximately normal, regardless of the shape of the population.

D) Because for any sample size n, it says the sampling distribution of the sample mean

is approximately normal.

ANSWER: C

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10. If all possible random samples of size n are taken from a population, and the mean of

each sample is determined, what can you say about the mean of the sample means?

A) It is larger than the population mean.

B) It is exactly the same as the population mean.

C) It is smaller than the population mean.

D) None of the above.

ANSWER: B

11. If the standard deviation of the sampling distribution of sample means is 5.0 for samples

of size 100, then the population standard deviation must be

A) 5.

B) 20.

C) 25

D) 50.

ANSWER: D

12. The amount of material used in making a custom seat cover for a sports car is normally

distributed. For a random sample of 16 seats, the mean amount of material used is 9.2

square feet, with a standard deviation of 2 square feet. Which of the following represents

a 99% confidence interval for the population mean amount of material used in a custom

sail?

A) 9.2 1.4735

B) 9.2 1.2525

C) 9.2 1.4605

D) 9.2 0.3684

ANSWER: A

13. Which of the following statements is correct?

A) An interval estimate describes a range of values that is likely not to include the actual

population parameter

B) An interval estimate is an estimate of the range for a sample statistic

C) An interval estimate is an estimate of the range of possible values for a population

parameter

D) All of the above

ANSWER: C

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14. Which of the following is not a property of the student's t distribution?

A) It is asymmetric.

B) Its shape is characterized by the degrees of freedom.

C) As the sample size increases, it gradually approaches the normal distribution.

D) All of the above are properties of the t distribution.

ANSWER: A

15. The larger the level of confidence (e.g., .99 versus .95) used in constructing a confidence

interval estimate of the population mean, the:

A) smaller the probability that the confidence interval will contain the population mean

B) narrower the confidence interval

C) smaller the value of 2/z

D) larger the confidence interval

ANSWER: A ?

16. A husband and wife, both are statisticians, decided to construct a 90% confidence

intervals for an unknown population mean. The husband selected a random sample of 50

observations while his wife's sample size was 80 observations. Which of the following is

true?

A) The wife's confidence interval has a greater degree of confidence.

B) The husband’s confidence interval has a greater degree of confidence.

C) The husband’s confidence interval is narrower.

D) The wife’s confidence interval is wider.

ANSWER: A

17. If a sample has observations and a 80% confidence estimate for is needed, the

appropriate t-score is:

A) 2.390

B) 1.671

C) 2.000

D) 1.296

ANSWER: D

18. If a sample of size 12 is selected, the value of A for the probability P(t A) = 0.01 is:

A) 2.718

B) -2.718

C) 2.681

D) -2.681

ANSWER: A

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19. The t- distribution approaches the normal distribution as the:

A) degrees of freedom increase

B) degrees of freedom decrease

C) sample size decreases

D) population size increases

ANSWER: A

20. Which of the following statement(s) is (are) correct about the z - distribution?

A) Has a mean of zero

B) Is a asymmetrical distribution

C) Is based on degrees of freedom

D) All of the above are correct

ANSWER: A

Part II: Question 2: Concepts (Choose A or B, not both) 10 marks

Question A: Describe the concept of stratified sampling. Illustrate the technique with an

example.

“The use of stratified sampling requires that a population be divided into homogeneous

groups called strata. Each stratum is then sampled according to certain specified criteria.” Under sampling with prior knowledge.

Divide population into strata.

Each strata is different.

Elements in the strata are the same.

Sample each strata to replicate the same socio-economic situation as the population.

Sampling is random within each strata.

Question B: Describe the concept of Unbiasness with respect to estimator properties.

On average, the value of the estimate should equal the population parameter being estimated.

If the average value of the estimator does not equal the actual parameter value, the estimator is a

biased estimator.

Ideally, an estimator has a bias of zero if it is said to be unbiased:

“An estimator is said to be unbiased if the expected value of the estimator is equal to the true value of the parameter being estimated.”

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Generally, if is a population parameter to be estimated,

and is an estimator where

( , , , ); X X Xn1 2

is said to be an unbiased estimator of if: E .

Example 1) )(XE under simple random sampling

(Topic 1); so X is an unbiased estimator.

The sample variance is an unbiased estimator of the population variance.

f

f

( )

(~)

f (

~)

E( ) E(~)

,~

Bias ~

[ (~) ] E

Two estimators:

~

is unbiased.

is biased.

Part III: Proofs

Question 3: Total marks:3

(i) Using the fact that the mean of the chi-squared distribution is (n-1), prove that .)( 22 sE

f ( )

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E s

E n

andn s

n sn

E sn

n

E s

2 2

2

22

2

2

2

2 2

2 2

1

1

11

1

1

Since

if you take the expectation:

E

( )

( )

( )

Part IV: SHORT ANSWER

Question 4: (6 marks)

Suppose we have two estimators of the population parameter :

32

2

ˆ

2ˆ

nV

nE

and

22

3

6~

2~

nV

nE

(i) Determine the bias, if any, of each estimator.

22 22)ˆ( nnEBias

33 22)~( nnEBias

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(ii) Determine the MSE. Which estimator is preferred?

4222 4)ˆ( nnbiasVMSE

6222 46)~( nnbiasVMSE

First estimator is preferred.

(iii) Determine if the estimators are consistent. Explain.

The bias and variance go to zero as n gets large.

Yes Consistent

Question 5: 11 marksConsider the following population of data: {9, 10, 11}.

(i) Determine the mean and variance of the population.

Total marks: 4

103/30)111093

1

2 2

1

2

1

21 1

N

XN

Xii

N

ii

N

( )

3

2

3

300302100

3

302

1 222

ixN

(ii) Determine the sampling distribution of the sample mean for a sample of size 2. Graph this distribution with

a simple bar graph.

Total marks: 4

X1,X2 X

9, 9 9

9, 10 9.5

9, 11 10

10, 9 9.5

10, 10 10

10, 11 10.5

Econ 246 Summer 2017 CRN#: 10991

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11, 9 10

11, 10 10.5

11, 11 11

X P( X )

9 1/9

9.5 2/9

10 3/9

10.5 2/9

11 1/9

(iii) Determine the variance of X ? Total Marks:3

(2/3) / 2=0.3333

P( X )

X

9 9.5 10 10.5 11

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Formulae

Central Location:

Population mean 1

Nxi

Grouped Population Mean

x f

f Nx f

i i

i

i i

1

Sample Mean Xn

X ii

n

1

1

Sample Mean for frequency distribution: Xn

X fi ii

n

1

1

Mean of the Sample Mean ( X ) E X X P XX i i

i

k

( ) ( )

1

where: i= 1,2,...,k, and k is the number of distinct possible values of X .

Dispersion:

Population variance 2 21

Nxi

(Grouped data 2 21

Nx fi i

Sample variance for frequency distribution: sn

x x fi i

2 21

1

( )

Sample variance sn

x xi

2 21

1

( )

Sample Standard Deviation s s 2

Variance of the Sample Mean X X

X

V X n X P X22

2 ( ) ( ) ( )

Standard Error of the mean:

X n n

2

.

Distributions:

Standard Normal:

ZX

( )

;

The Standardization of X:

ZX

n

t-distribution

tX

sn

; Chi-square distribution

n

s n s

1

2

2

2

2

2

1( ) ( )

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