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Hypothesis Testing,Regression Analysis&Chi-square Test

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.

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