Statistics for business and economics

Since 1987 almost all new buildings are using electricity or natural gas with natural gas being the clear leader... Management should be encouraged by the fact that steps taken in 2003 r

modified 2/16/2010 EXCERPTS FROM:

Solutions Manual to Accompany

Statistics for Business

Rochester Institute of Technology

The material from which this was excerpted is copyrighted by

SOUTH-WESTERN

Contents

1 Data and Statistics 1

2 Descriptive Statistics: Tabular and Graphical Methods 2

3 Descriptive Statistics: Numerical Methods 5

4 Introduction to Probability 8

5 Discrete Probability Distributions 11

6 Continuous Probability Distributions 13

7 Sampling and Sampling Distributions 15

8 Interval Estimation 17

9 Hypothesis Testing 18

10 Statistical Inference about Means and Proportions with Two populations 22

14 Simple Linear regression 25

15 Multiple Regression 30

16 Regression Analysis: Model Building 35

21 Decision Analysis 37

1 Data and Statistics

12 a The population is all visitors coming to the state of Hawaii

b Since airline flights carry the vast majority of visitors to the state, the use of questionnaires for passengers during incoming flights is a good way to reach this population The questionnaire actually appears on the back of a mandatory plants and animals declaration form that passengers must complete during the incoming flight A large percentage of passengers complete the visitor information questionnaire

c Questions 1 and 4 provide quantitative data indicating the number of visits and the number of days

in Hawaii Questions 2 and 3 provide qualitative data indicating the categories of reason for the trip and where the visitor plans to stay

21 a The two populations are the population of women whose mothers took the drug DES during

pregnancy and the population of women whose mothers did not take the drug DES during

pregnancy

b It was a survey

c 63 / 3.980 = 15.8 women out of each 1000 developed tissue abnormalities

d The article reported “twice” as many abnormalities in the women whose mothers had taken DES during pregnancy Thus, a rough estimate would be 15.8/2 = 7.9 abnormalities per 1000 women

whose mothers had not taken DES during pregnancy

e In many situations, disease occurrences are rare and affect only a small portion of the population Large samples are needed to collect data on a reasonable number of cases where the disease exists

2 Descriptive Statistics: Tabular and Graphical Methods

5

13

1230

100.0

100.0Total

0.0

16.7

83.3100.0

1 2

Total

d Category A values for x are always associated with category 1 values for y Category B values for x are usually associated with category 1 values for y Category C values for x are usually associated with category 2 values for y

e Observations from the column percentages crosstabulation

For those buildings using electricity, the percentage has not changed greatly over the years For the buildings using natural gas, the majority were constructed in 1973 or before; the second largest percentage was constructed in 1987-1991 Most of the buildings using oil were constructed in 1973

or before All of the buildings using propane are older

Observations from the row percentages crosstabulation

Most of the buildings in the CG&E service area use electricity or natural gas In the period 1973 or before most used natural gas From 1974-1986, it is fairly evenly divided between electricity and natural gas Since 1987 almost all new buildings are using electricity or natural gas with natural gas being the clear leader

3 Descriptive Statistics: Numerical Methods

20

i

x x

c Mode = $167 San Francisco and New Orleans

s

Approximately two standard deviations above the mean Approximately 95% of the scores are within two standard deviations Thus, half of (100-95), or 2.5%, of the games should have a winning score of more than 90 points

10

i x x

Σ

= = =-1-0.500.51

i x

i y

xy xy

HT

HT

HT

HT

(H,H,H)(H,H,T)(H,T,H)(H,T,T)(T,H,H)(T,H,T)(T,T,H)(T,T,T)1st Toss 2nd Toss 3rd Toss

b Let: H be head and T be tail

c The outcomes are equally likely, so the probability of each outcomes is 1/8

21 a Use the relative frequency method Divide by the total adult population of 227.6 million

b Let A = 4- or 5-star rating

13 funds were rated 3-star of less; thus, 25 – 13 = 12 funds must be 4-star or 5-star

P(A) = 12/25 = 48

c 7 Domestic Equity funds were rated 4-star and 2 were rated 5-star Thus, 9 funds were Domestic Equity funds and were rated 4-star or 5-star

d P(D ∪ A) = P(D) + P(A) - P(D ∩ A)

= 64 + 48 - 36 = 76

P = rented a car for personal reasons

P

∩

= = =

c No P(A | B) ≠ P(A); ∴ the events, although mutually exclusive, are not independent

34 a Let O = flight arrives on time

(S)

P P

c

.2282(O )

P P

43 Let: S = small car

F = accident leads to fatality for vehicle occupant

of Bayes Theorem provides:

Events

Prior Probabilities

Conditional Probabilities

Joint Probabilities

Posterior Probabilities

b Let S = Soil test results

.304

P(F D) 67

3 1 30 15.40 60

The revised (posterior) probability that the visitor is female is 67

We should display the offer that appeals to female visitors

5 Discrete Probability Distributions

b It may assume any positive value: x > 0

14 a f (200) = 1 - f (-100) - f (0) - f (50) - f (100) - f (150)

= 1 - 95 = 05 This is the probability MRA will have a $200,000 profit

e f

5!

e f

−

= =

use the binomial probability distribution

a n = 5

5(0) (0.01) (0.99) 0.9510

d No, the probability of finding one or more items in the sample defective when only 1% of the items

in the population are defective is small (only 0490) I would consider it likely that more than 1% of the items are defective

6 Continuous Probability Distributions

b .683 since 45 and 55 are within plus or minus 1 standard deviation from the mean of 50 (Use the

table or see characteristic 7a of the normal distribution)

.954 since 40 and 60 are within plus or minus 2 standard

table or see characteristic 7b of the normal distribution)

The z value corresponding to a cumulative probability of 9948 is z = 2.56

e The area to the left of z is 1 - 6915 = 3085 So z = -.50

4

Expected number of defects = 1000(.3174) = 317.4

b P(defect) = 1 - P(9.85 ≤ x ≤ 10.15) = 1 - P(-3 ≤ z

Expected number of defects = 1000(.0026) = 2.6

7 Sampling and Sampling Distributions

n

μσ

.18 15

1.03.0292

P(.12 ≤p ≤ 18) = P(-1.03 ≤ z ≤ 1.03) = 8485 - 1515 =.6970

8 Interval Estimation

A larger sample size would be needed to reduce the margin of error to $150 or less Section 8.3 can

be used to show that the sample size would need to be increased to n = 62

d As the confidence level increases, there is a larger margin of error and a wider confidence interval

d Skewness = 1.0062, data are skewed to the right

Regardless of skewness, this is a pretty small data set Consider using a larger sample next time

z p p n

E

9 Hypothesis Testing

b We are not able to conclude that the manager’s claim is wrong

c The manager’s claim can be rejected We can conclude that μ > 600

b There is no statistical evidence that the new bonus plan increases sales volume

c The research hypothesis that μ > 14 is supported We can conclude that the new bonus plan increases the mean sales volume

plan when it does not help

implementing a plan that would increase sales

1.48/ 6 / 40

x z

n

μσ

− −

c p-value > 01, do not reject H0

μ

− −

mean CNN viewing audience

d The sample mean of 612 thousand viewers is encouraging but not conclusive for the sample of 40 days Recommend additional viewer audience data A larger sample should help clarify the situation for CNN

n

Σ

c ( )2

.5161

e p-value > 05; do not reject H0 No reason to change from the 2 hours for cost estimating purposes

.68 75 2.80(1 ) 75(1 75)

300

p p z

p-value ≤ 05; Reject H0

.75(1 75)300

p-value > 05; Do not reject H0

c .70 75 2.00

.75(1 75)300

− −

p-value ≤ 05; Reject H0

d .77 75 80

.75(1 75)300

−

Using normal table with z = 80: p-value =.7881

p-value > 05; Do not reject H0

1532

p p z

Conclude that there has been a significant increase in the intent to watch the TV programs

c These studies help companies and advertising firms evaluate the impact and benefit of commercials

50

c 0

.48 30

2.78(1 ) 30(1 30)

50

p p z

p-value .01; reject H≤ 0

We would conclude that the proportion of stocks going up on the NYSE is not 30% This would suggest not using the proportion of DJIA stocks going up on a daily basis as a predictor of the proportion of NYSE stocks going up on that day

6883 5980

x

Degrees of freedom = n – 1 = 39

c We should conclude that Medicare spending per enrollee in Indianapolis is less than the national average

d Using the critical value approach we would:

n

Σ

.04441

p-value > 05; do not reject H0

There is not a statistically significant difference between the National mean price per gallon and the mean price per gallon in the Lower Atlantic states

400

p= =

c 0

.2025 24

1.76(1 ) 24(1 24)

400

p p z

Using normal table with z = -1.76: p-value =.0392

p-value ≤ 05; reject H0

The proportion of workers not required to contribute to their company sponsored health care plan has declined There seems to be a trend toward companies requiring employees to share the cost of health care benefits

10 Statistical Inference about Means and Proportions with Two populations

e Percentage reduction: 6/172 = 3.5% Management should be encouraged by the fact that steps taken

in 2003 reduced the population mean duration of baseball games However, the statistical analysis shows that the reduction in the mean duration is only 3.5% The interval estimate shows the reduction in the population mean is 1.5 minutes (.9%) to 10.5 minutes (6.1%) Additional data collected by the end of the 2003 season would provide a more precise estimate In any case, most likely the issue will continue in future years It is expected that major league baseball would prefer that additional steps be taken to further reduce the mean duration of games

d d s

d d

d t

c Groceries has the higher mean annual expenditure by an estimated $850

i

d d

d d s

weekly usage for the two media

≤

15

i TV

x x

x x n

∑

Radio has greater usage

Professional golfers have the better putting accuracy

The confidence interval shows that professional golfers make from 2% to 10% more 6-foot putts

than the best amateur golfers

= = − p-value = 0025

14 Simple Linear regression

b There appears to be a positive linear relationship between x = features rating and y = PCW World

i

x x y y b

44 a/b The scatter diagram shows a linear relationship between the two variables

c The Minitab output is shown below:

The regression equation is

Predicted Values for New Observations

New Obs Fit SE Fit 95.0% CI 95.0% PI

1 17.59 2.51 ( 12.27, 22.90) ( 5.94, 29.23)

2 28.26 1.42 ( 25.26, 31.26) ( 17.47, 39.05) Values of Predictors for New Observations

New Obs Vacancy%

1 25.0

2 11.3

f The 95% confidence interval is 12.27 to 22.90 or $12.27 to $22.90

g The 95% prediction interval is 17.47 to 39.05 or $17.47 to $39.05

47 a Let x = advertising expenditures and y = revenue

Using Excel or Minitab, the p-value corresponding to F = 11.15 is 0206

and y but the fit is not very good

60 a

11601180120012201240126012801300

DJIA

b The Minitab output is shown below:

The regression equation is

d With R-Sq = 95.6%, the estimated regression equation provided an excellent fit

f The DJIA is not that far beyond the range of the data With the excellent fit provided by the

estimated regression equation, we should not be too concerned about using the estimated regression equation to predict the S&P500

15 Multiple Regression

5 a The Minitab output is shown below:

The regression equation is

Total 7 25.500

b The Minitab output is shown below:

The regression equation is

Revenue = 83.2 + 2.29 TVAdv + 1.30 NewsAdv

Predictor Coef SE Coef T P

Total 7 25.500

c No, it is 1.60 in part (a) and 2.29 above In part (b) it represents the marginal change in revenue due

to an increase in television advertising with newspaper advertising held constant

d Revenue = 83.2 + 2.29(3.5) + 1.30(1.8) = $93.56 or $93,560

7 a The Minitab output is shown below:

The regression equation is

Source DF SS MS F P

Regression 1 66.343 66.343 9.87 0.014

Residual Error 8 53.757 6.720

Total 9 120.100

b The Minitab output is shown below:

The regression equation is

PCW Rating = 40.0 + 0.113 Performance + 0.382 Features

Predictor Coef SE Coef T P

7 a The Minitab output is shown below:

The regression equation is

Price = 356 - 0.0987 Capacity + 123 Comfort

Predictor Coef SE Coef T P

to a 1 unit change in the comfort rating with the capacity held constant

NOTE: These answers seem to imply that a variable whose p-value is above alpha should be dropped

THAT IS NOT NECESSARILY TRUE!

The Minitab output is shown below:

Fit Stdev.Fit 95% C.I 95% P.I

93.588 0.291 ( 92.840, 94.335) ( 91.774, 95.401)

b Confidence interval estimate: 92.840 to 94.335 or $92,840 to $94,335

c Prediction interval estimate: 91.774 to 95.401 or $91,774 to $95,401

34 a $15,300

b Estimate of sales = 10.1 - 4.2(2) + 6.8(8) + 15.3(0) = 56.1 or $56,100

c Estimate of sales = 10.1 - 4.2(1) + 6.8(3) + 15.3(1) = 41.6 or $41,600

35 a Let Type = 0 if a mechanical repair

Type = 1 if an electrical repair The Minitab output is shown below:

The regression equation is

Total 9 10.476

b The estimated regression equation did not provide a good fit In fact, the p-value of 408 shows that

the relationship is not significant for any reasonable value of α

c Person = 0 if Bob Jones performed the service and Person = 1 if Dave Newton performed the service The Minitab output is shown below:

The regression equation is

Total 9 10.4760

d We see that 61.1% of the variability in repair time has been explained by the repair person that performed the service; an acceptable, but not good, fit

36 a The Minitab output is shown below:

The regression equation is

Time = 1.86 + 0.291 Months + 1.10 Type - 0.609 Person

Predictor Coef SE Coef T P

Total 9 10.4760

significant

statistically significant Person is highly correlated with Months (the sample correlation coefficient

is -.691); thus, once the effect of Months has been accounted for, Person will not add much to the model

42 a The Minitab output is shown below:

The regression equation is

Speed = 71.3 + 0.107 Price + 0.0845 Horsepwr

Predictor Coef SE Coef T P

X denotes an observation whose X value gives it large influence

b The standardized residual plot is shown below There appears to be a very unusual trend in the

110 105

100 95