# Hypothesis Testing,Regression Analysis&Chi-square Test

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10.30 In Dallas, some fire trucks were painted yellow (instead of red) to heighten their visibility. During

a test period, the fleet of red fire trucks made 153,348 runs and had 20 accidents, while the fleet of

yellow fire trucks made 135,035 runs and had 4 accidents. At α = .01, did the yellow fire trucks

have a significantly lower accident rate? (a) State the hypotheses. (b) State the decision rule and

sketch it. (c) Find the sample proportions and z test statistic. (d) Make a decision. (e) Find the

p-value and interpret it. (f ) If statistically significant, do you think the difference is large enough to

be important? If so, to whom, and why? (g) Is the normality assumption fulfilled? Explain.

Source: The Wall Street Journal, June 26, 1995, p. B1.

Accident Rate for Dallas Fire Trucks

Statistic Red Fire Trucks Yellow Fire Trucks

Number of accidents x1 = 20 accidents x2 = 4 accidents

Number of fire runs n1 = 153,348 runs n2 = 135,035 runs

10.44 Does lovastatin (a cholesterol-lowering drug) reduce the risk of heart attack? In a Texas study,

researchers gave lovastatin to 2,325 people and an inactive substitute to 2,081 people (average age

58). After 5 years, 57 of the lovastatin group had suffered a heart attack, compared with 97 for the

inactive pill. (a) State the appropriate hypotheses. (b) Obtain a test statistic and p-value. Interpret

Doane−Seward: Applied

Statistics in Business and

Economics

10. Two−Sample

Hypothesis Tests

Text © The McGraw−Hill

Companies, 2007

Chapter 10 Two-Sample Hypothesis Tests 431

the results at α = .01. (c) Is normality assured? (d) Is the difference large enough to be important?

(e) What else would medical researchers need to know before prescribing this drug widely? (Data

are from Science News 153 [May 30, 1998], p. 343.)

10.46 To test the hypothesis that students who finish an exam first get better grades, Professor Hardtack

kept track of the order in which papers were handed in. The first 25 papers showed a mean score of

77.1 with a standard deviation of 19.6, while the last 24 papers handed in showed a mean score of

69.3 with a standard deviation of 24.9. Is this a significant difference at α = .05? (a) State the

hypotheses for a right-tailed test. (b) Obtain a test statistic and p-value assuming equal variances.

Interpret these results. (c) Is the difference in mean scores large enough to be important? (d) Is it reasonable

to assume equal variances? (e) Carry out a formal test for equal variances at α = .05, showing

all steps clearly.

10.56 A sample of 25 concession stand purchases at the October 22 matinee of Bride of Chucky showed

a mean purchase of $5.29 with a standard deviation of $3.02. For the October 26 evening showing

of the same movie, for a sample of 25 purchases the mean was $5.12 with a standard deviation of

$2.14. The means appear to be very close, but not the variances. At α = .05, is there a difference

in variances? Show all steps clearly, including an illustration of the decision rule. (Data are from

a project by statistics students Kim Dyer, Amy Pease, and Lyndsey Smith.)

11.24 In a bumper test, three types of autos were deliberately crashed into a barrier at 5 mph, and the

resulting damage (in dollars) was estimated. Five test vehicles of each type were crashed, with

the results shown below. Research question: Are the mean crash damages the same for these three

vehicles? Crash1

Goliath Varmint Weasel

1,600 1,290 1,090

760 1,400 2,100

880 1,390 1,830

1,950 1,850 1,250

1,220 950 1,920

9.56 A coin was flipped 60 times and came up heads 38 times. (a) At the .10 level of significance, is the

coin biased toward heads? Show your decision rule and calculations. (b) Calculate a p-value and

interpret it.

12.48 In the following regression, X = weekly pay, Y = income tax withheld, and n = 35 McDonald's

employees. (a) Write the fitted regression equation. (b) State the degrees of freedom for a twotailed

test for zero slope, and use Appendix D to find the critical value at α = .05. (c) What is your

conclusion about the slope? (d) Interpret the 95 percent confidence limits for the slope. (e) Verify

that F = t2 for the slope. (f) In your own words, describe the fit of this regression

R2 0.202

Std. Error 6.816

n 35

ANOVA table

__________________________________________________________________

Source SS df MS F p-value

Regression 387.6959 1 387.6959 8.35 . 0068

Residual 1,533.0614 33 46.4564

___________________________________________________________________

Total 1,920.7573 34

Regression output confidence interval

_________________________________________________________________________________________________

Variables coefficients std. error t (df = 33) p-value 95% lower 95% upper

___________________________________________________________________________________________________

Intercept 30.7963 6.4078 4.806 .0000 17.7595 43.8331

Slope 0.0343 0.0119 2.889 .0068 0.0101 0.0584

12.50 In the following regression, X = total assets ($ billions), Y = total revenue ($ billions), and n = 64

large banks. (a) Write the fitted regression equation. (b) State the degrees of freedom for a twotailed

test for zero slope, and use Appendix D to find the critical value at α = .05. (c) What is your

conclusion about the slope? (d) Interpret the 95 percent confidence limits for the slope. (e) Verify

that F = t2 for the slope. (f) In your own words, describe the fit of this regression.

R2 0.519

Std. Error 6.977

n 64

ANOVA table

__________________________________________________________________

Source SS df MS F p-value

Regression 3,260.0981 1 3,260.0981 66.97 1.90E-11

Residual 3,018.3339 62 48.6828

___________________________________________________________________

Total 6,278.4320 63

Regression output confidence interval

_________________________________________________________________________________________________

Variables coefficients std. error t (df = 33) p-value 95% lower 95% upper

___________________________________________________________________________________________________

Intercept 6.5763 1.9254 3.416 .0011 2.7275 10.4252

X1 0.0452 0.0055 8.183 1.90E-11 0.0342 0.0563

 

13.30 A researcher used stepwise regression to create regression models to predict BirthRate (births per

1,000) using five predictors: LifeExp (life expectancy in years), InfMort (infant mortality rate),

Density (population density per square kilometer), GDPCap (Gross Domestic Product per capita),

and Literate (literacy percent). Interpret these results.

Regression Analysis-Stepwise Selection (best model of each size)

153 observations

Birth Rate is the dependent variable

p-values for the coefficients

Nvar LifeExp InfMort Density GDPCap Literate s Adj R2 R2

1 .0000 6.318 .722 .724

2 .0000 .0000 5.334 .802 .805

3 .0000 .0242 .0000 5.261 .807 .811

4 .5764 .0000 .0311 .0000 5.273 .806 .812

5 .5937 .0000 .6289 .0440 .0000 5.287 .805 .812

13.32 An expert witness in a case of alleged racial discrimination in a state university school of nursing

introduced a regression of the determinants of Salary of each professor for each year during an

8-year period (n = 423) with the following results, with dependent variable Year (year in which

the salary was observed) and predictors YearHire (year when the individual was hired), Race (1 if

individual is black, 0 otherwise), and Rank (1 if individual is an assistant professor, 0 otherwise).

Interpret these results.

Variable Coefficient t p

Intercept −3,816,521 −29.4 . 000

Year 1,948 29.8 .000

YearHire −826 − 5.5 .000

Race −2,093 −4.3 .000

Rank −6,438 −22.3 .000

R2 = 0.811 R2adj = 0.809 s = 3,318

 

14.16 (a) Plot the data on U.S. general aviation shipments. (b) Describe the pattern and discuss possible

causes. (c) Would a fitted trend be helpful? Explain. (d) Make a similar graph for 1992-2003 only.

Would a fitted trend be helpful in making a prediction for 2004? (e) Fit a trend model of your

choice to the 1992-2003 data. (f) Make a forecast for 2004, using either the fitted trend model or

a judgment forecast. Why is it best to ignore earlier years in this data set?

U.S. Manufactured General Aviation Shipments, 1966-2003

Year Planes Year Planes Year Planes Year Planes

1966 15,587 1976 15,451 1986 1,495 1996 1,053

1967 13,484 1977 16,904 1987 1,085 1997 1,482

1968 13,556 1978 17,811 1988 1,143 1998 2,115

1969 12,407 1979 17,048 1989 1,535 1999 2,421

1970 7,277 1980 11,877 1990 1,134 2000 2,714

1971 7,346 1981 9,457 1991 1,021 2001 2,538

1972 9,774 1982 4,266 1992 856 2002 2,169

1973 13,646 1983 2,691 1993 870 2003 2,090

1974 14,166 1984 2,431 1994 881

1975 14,056 1985 2,029 1995 1,028

15.18 In a three-digit lottery, each of the three digits is supposed to have the same probability of occurrence

(counting initial blanks as zeros, e.g., 32 is treated as 032). The table shows the frequency

of occurrence of each digit for 90 consecutive daily three-digit drawings. (a) Make a bar chart and

describe it. (b) Calculate expected frequencies for each class. (c) Perform the chi-square test for a

uniform distribution. At α = .05, can you reject the hypothesis that the digits are from a uniform

population? Lottery3

Digit Frequency

0 33

1 17

2 25

3 30

4 31

5 28

6 24

7 25

8 32

9 25

Total 270

15.22 A student team examined parked cars in four different suburban shopping malls. One hundred vehicles

were examined in each location. Research question: At α = .05, does vehicle type vary by

mall location? (Data are from a project by MBA students Steve Bennett, Alicia Morais, Steve

Olson, and Greg Corda.) Vehicles

Vehicle Type Somerset Oakland Great Lakes Jamestown Row Total

Car 44 49 36 64 193

Minivan 21 15 18 13 67

Full-sized Van 2 3 3 2 10

SUV 19 27 26 12 84

Truck 14 6 17 9 46

Col Total 100 1 00 100 100 400

15.24 High levels of cockpit noise in an aircraft can damage the hearing of pilots who are exposed to this

hazard for many hours. A Boeing 727 co-pilot collected 61 noise observations using a handheld

sound meter. Noise level is defined as "Low" (under 88 decibels), "Medium" (88 to 91 decibels),

or "High" (92 decibels or more). There are three flight phases (Climb, Cruise, Descent). Research

question: At α = .05, is the cockpit noise level independent of flight phase? (Data are from

Capt. Robert E. Hartl, retired.)

Noise Level Climb Cruise Descent Row Total

Low 6 2 6 14

Medium 18 3 8 29

High 1 3 14 18

Col Total 25 8 28 61

15.28 Can people really identify their favorite brand of cola? Volunteers tasted Coca-Cola Classic,

Pepsi, Diet Coke, and Diet Pepsi, with the results shown below. Research question: At α = .05, is

the correctness of the prediction different for the two types of cola drinkers? Could you identify

your favorite brand in this kind of test? Since it is a 2 × 2 table, try also a two-tailed two-sample

z test for π1 = π2 (see Chapter 10) and verify that z2 is the same as your chi-square statistic.Which

test do you prefer? Why? (Data are from Consumer Reports 56, no. 8 [August 1991], p. 519.)

Correct? Regular Cola Diet Cola Row Total

Yes, got it right 7 7 14

No, got it wrong 12 20 32

Col Total 19 27 46

#### Solution Summary

The solution provides step by step method for the calculation of test statistic of population mean, population proportion, chi-square and ANOVA. The solution also provides step by step method for the calculation of regression analysis. Formula for the calculation and Interpretations of the results are also included. Interactive excel sheet is included. The user can edit the inputs and obtain the complete results for a new set of data.