CHAPTER 7 RESERVE PROBLEMS

The following problems have been reserved for your use in assignments and testing and do not

appear in student versions of the text.

Reserve Problems Chapter 7 Section 1 Problem 1

A random sample of 140 in size is taken from a population with a mean of 1540 and unknown

variance. The sample variance was found out to be 130.

(a) Find the point estimate of the population variance.

(b) Find the mean of the sampling distribution of the sample mean.

SOLUTION

Reserve Problems Chapter 7 Section 1 Problem 2

Service time for a customer coming through a checkout counter in a retail store is a random

variable with the mean of 4.0 minutes and standard deviation of 1.5 minutes. Suppose that the

distribution of service time is fairly close to a normal distribution. Suppose there are two

counters in a store,

141n=

customers in the first line and

251n=

customers in the second line.

Find the probability that the difference between the mean service time for the shorter line

1

X

and

the mean service time for the longer one

2

X

is more than 0.4 minutes. Assume that the service

times for each customer can be regarded as independent random variables.

SOLUTION

Reserve Problems Chapter 7 Section 1 Problem 3

Suppose that the random variable X has the discrete uniform distribution

( )

1/ 4, 3, 4,5, 6.

0, .

x

fx otherwise

=

=

A random sample of

20n=

is selected from this distribution. Find the probability that the

sample mean is greater than 4.8.

SOLUTION

Reserve Problems Chapter 7 Section 1 Problem 4

In order to find out the defect rate of the manufactured components a random sample of

160n=

was selected. Four specimens were found to be defective. Estimate the proportion of defective

components in the population.

SOLUTION

Reserve Problems Chapter 7 Section 2 Problem 1

Consider a normal population with the mean of 40 and standard deviation of 10. A random

sample of

10n=

was selected:

39.2, 45.7, 27.4, 25.9, 25.1, 46.3, 42.9, 49.0, 40.6, 47.0.

Find the probability that the point estimate of the population mean based on the second sample

of

20n=

would be more accurate.

SOLUTION

138.91X=

,

Reserve Problems Chapter 7 Section 2 Problem 2

A normal population has the mean of 20 and the variance of 100. A random sample of size

69n=

is selected.

(a) Find the standard deviation of the sample mean.

(b) How large must the sample be if you want to halve the standard deviation of the sample

mean?

SOLUTION

(a)

Reserve Problems Chapter 7 Section 2 Problem 3

The lamps of type A have the average lifetime of 2000 hours with the variance of 3500. The

lamps of type B have the average lifetime of 2100 hours with the same variance. A random

sample of

100n=

lamps of each type is selected. Let

A

X

and

B

X

be the two sample means.

Find the probability that

90

BA

XX−

.

SOLUTION

Reserve Problems Chapter 7 Section 2 Problem 4

Patients arriving at a hospital emergency department present a variety of symptoms and

complaints. The following data were collected during one weekend night shift (11:00 P.M. to

7:00 A.M.):

Chest pain

7

Difficulty breathing

7

Numbness in extremities

3

Broken bones

11

Abrasions

16

Cuts

21

Stab wounds

9

Gunshot wounds

4

Blunt force trauma

10

Fainting, loss of consciousness

5

Other

9

Estimate the proportion of patients who arrive at this emergency department experiencing chest

pain.

SOLUTION

Reserve Problems Chapter 7 Section 2 Problem 5

A consumer electronics company is comparing the brightness of two different types of picture

tubes for use in its television sets. Tube type A has mean brightness of 100 and standard

deviation of 16, and tube type B has unknown mean brightness, but the standard deviation is

assumed to be identical to that for type A. A random sample of

25n=

tubes of each type is

selected, and

BA

XX

−

is computed. If

B

equals or exceeds

A

, the manufacturer would like to

adopt type B for use. The observed difference is

3.0

BA

xx−=

.

What is the probability that

B

X

exceeds

A

X

by 3.0 or more if

B

and

A

are equal?

Is there strong evidence that

B

is greater than

A

?

SOLUTION

Reserve Problems Chapter 7 Section 2 Problem 6

Consider a Weibull distribution with shape parameter 1.5 and scale parameter 2.0.

2.39

1.14

0.56

3.48

1.57

1.53

2.78

3.01

0.55

1.29

2.22

0.64

0.49

0.56

0.11

2.59

1.01

0.43

1.62

3.43

5.50

1.55

0.59

0.61

0.68

0.54

0.99

1.48

2.23

6.25

1.23

2.52

1.27

1.01

6.26

2.43

0.27

1.33

1.27

2.72

0.26

3.59

1.91

0.02

2.94

2.47

0.85

1.58

2.40

2.06

Select the correct histogram.

A

B

C

Normal probability plot: Does it look very much like a normal distribution?

The table constructed by drawing 10 samples of size n=5 from this distribution.

Compute the sample average from each sample.

Obs

Mean

1

2.72

1.27

1.01

6.26

0.55

2.36 ? 0.01

2

0.99

2.59

5.50

1.27

0.64

2.20 ? 0.01

3

6.25

0.61

2.43

6.26

0.68

3.25 ? 0.01

4

0.26

1.29

1.23

1.14

1.57

1.10 ? 0.01

5

1.57

1.01

2.43

0.26

1.58

1.37 ? 0.01

6

1.62

2.72

2.78

2.39

2.94

2.49 ? 0.01

7

0.56

2.47

1.91

0.99

0.27

1.24 ? 0.01

8

2.06

0.59

2.59

1.14

0.54

1.38 ? 0.01

9

3.48

1.55

1.14

3.43

0.68

2.06 ? 0.01

10

1.23

1.01

2.47

0.56

3.59

1.77 ? 0.01

Select the correct normal probability plot of the sample averages.

A

B

C

Do the sample averages seem to be normally distributed?

SOLUTION

The histogram of the data:

Obs

Mean

1

2.72

1.27

1.01

6.26

0.55

2.36

Normal probability plot of the sample averages.

Reserve Problems Chapter 7 Section 3 Problem 1

We have two unbiased estimators:

ˆX

=

, and

22

ˆS

=

. Suppose that X is a random variable

with mean

and variance

2

. There is a random sample of size

5n=

from a population

represented by

1 2 3 4 5

: 28, 27, 34, 21, 38X x x x x x= = = = =

. Find a point estimate of

and

2

.

Use unbiased estimators.

SOLUTION

The unbiased estimator of

is the sample mean:

Reserve Problems Chapter 7 Section 3 Problem 2

Suppose that a random variable X has continuous uniform distribution on

1, a

, where a is an

unknown parameter. We have a random sample of 15 in size from a population represented by X:

4.5, 1.3, 8.6, 6.4, 7.4, 4.3, 7.2, 1.6, 4.4, 2.0, 8.4, 6.5, 7.3, 3.4, 7.4..

Find a point estimate of a. Use an unbiased estimator.

SOLUTION

The expected value of X for continuous uniform distribution:

Reserve Problems Chapter 7 Section 3 Problem 3

At the candle factory, 10 out of 100 produced candles in the sample are defective. Find a point

estimate of the probability that a fault free candle is produced.

SOLUTION

Reserve Problems Chapter 7 Section 3 Problem 4

The number of passengers for a particular bus route was counted, and data for 3 weeks are

shown in the table:

Mon

Tue

Wed

Thu

Fri

Sat

Sun

Week 1

29

33

26

28

30

39

31

Week 2

28

31

29

37

28

37

25

Week 3

33

37

30

28

25

39

28

(a) Calculate a point estimate of the mean number of passengers and its standard error. Consider

the unbiased estimator

ˆX

=

.

(b) Calculate a point estimate of the mean number of passengers and its standard error on

weekdays and weekends separately.

SOLUTION

(a)

A point estimate of the mean number:

Reserve Problems Chapter 7 Section 3 Problem 5

The diameter of the cylinder is 52 cm, its height was measured 7 times. Measurements are as

follows:

58.3, 58.1, 57.9, 57.7, 58.2, 57.8, 58.1cm. Calculate a point estimate of the cylinder volume.

Consider the unbiased estimator

X

. Use

3.14

=

.

SOLUTION

Reserve Problems Chapter 7 Section 3 Problem 6

A boy threw a ball 25 times. Kinetic energy of the ball is as follows:

25.66, 30.71, 18.32, 16.77, 24.05, 19.49, 14.22, 20.03, 14.21, 28.43, 28.03, 22.07, 29.48, 14.91,

17.12, 15.12, 27.97, 23.02, 23.37, 27.55, 23.48, 19.90, 15.78, 28.88, 29.71 J. Calculate a point

estimate of the proportion of all ball throws whose energy deviation from the mean is larger than

the standard deviation.

SOLUTION

Reserve Problems Chapter 7 Section 3 Problem 7

Suppose that

1

ˆ

and

2

ˆ

are estimators of the parameter

. We know that

( )

1

ˆ

E

=

,

( )

22

ˆ/E

=

,

( )

10

ˆ1V=

,

( )

2

ˆ4V=

.

Which estimator is better for unbiasedness?

Under which conditions

2

ˆ

is more efficient than

1

ˆ

?

SOLUTION

Reserve Problems Chapter 7 Section 3 Problem 8

Let three random samples of sizes

118n=

,

210n=

, and

38n=

be taken from a population with

mean

and variance

2

. Let

2

1

S

,

2

2

S

, and

2

3

S

be the sample variances.

Is

( )

2 2 2 2

1 2 3

18 10 8 / 36S S S S= + +

biased or unbiased estimator of

2

?

SOLUTION

Reserve Problems Chapter 7 Section 3 Problem 9

Suppose that the random variable X has a lognormal distribution with parameters

1.5

=

and

0.8

=

. A sample of size

10n=

is drawn from this distribution. In order to find the standard

error of the sample median of this distribution with the bootstrap method

5

B

n=

bootstrap

samples are generated.

Sample

1

7.01

3.58

7.04

3.58

7.86

14.68

3.01

8.56

7.66

10.84

2

4.75

14.66

4.76

7.66

5.29

4.48

4.98

6.07

6.37

1.50

3

23.72

2.80

6.03

5.89

3.58

2.22

6.50

1.52

3.19

9.59

4

1.50

4.75

5.63

1.87

5.27

1.83

7.66

5.63

4.55

4.20

5

6.98

10.21

1.50

6.03

4.97

3.19

10.86

6.37

6.33

4.98

Find the standard error of the sample median of this distribution with the bootstrap method.

SOLUTION

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