Workshop Statistics: Discovery with Data


Brief Answers to Selected In-Class Activities

Compiled by Kathy L. Clawson, Dickinson College class of 1998






This Web page contains brief answers to selected in-class activities from Workshop Statistics. These answers address questions which have numerical answers or otherwise short answers. You should note that our answers here are often more succinct than we would prefer students to be. In the interest of brevity we also break the admonition that we expect of students to write in complete sentences and to relate all comments to the context of the data. We do not supply answers to questions calling for explanations, interpretations, guesses, predictions, or expectations on the part of students. For obvious reasons we do not supply answers to questions that ask for the analysis of data collected in class.

Despite this lack of thoroughness, we hope that this guide will prove useful to instructors. For a more complete perspective, you should use these answers in conjunction with the Guide for Instructors, which offers suggestions on what to look for and to emphasize in each in-class activity.

This page contains none of the graphics, but you can also access a with graphics version.

This document is fairly lengthy, and this should be kept in mind if a print out is to be made.

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List of In-Class Activities






1-1: Types of Variables

(a) gender: categorical (binary)
politcal identification: categorical
penny question: categorical (binary)
value of statistics: measurement
number of states: measurement
number of countries: measurement
Europe?: categorical (binary)
WDW?: categorical (binary)
letters per word: measurement

(b) categorical

(c) measurement

1-2: Penny Thoughts

(a) - (d) These answers depend on class results.

1-3: Value of Statistics

TABLE - These answers depend on class results.

1-4: Students' Travels

(a) - (d) These answers depend on class results.

1-5: Gender of Physicians

(a)
Specialty -- Percentage Women
Aerospace medicine -- 5.93%
Allergy and immunology --16.04%
Anesthesiology -- 18.37%
Cardiovascular disease -- 5.55%
Child psychiatry -- 35.45%
Colon/rectal surgery -- 3.34%
Dermatology -- 24.09%
Diagnostic radiology --16.80%
Emergency medicine -- 15.25%
Family practice -- 19.05%
Forensic pathology -- 23.17%
Gastroenterology -- 6.39%
General practice -- 11.24%
General preventive medicine -- 25.43%
General surgery -- 7.22%
Internal medicine -- 21.79%
Neurological surgery -- 3.24%
Neurology -- 16.75%
Nuclear medicine -- 15.74%
Obstetrics/gynecology -- 25.61%
Occupational medicine -- 11.55%
Opthamology -- 10.60%
Orthopedic surgery -- 2.49%
Otolaryngology -- 5.86%
Pathology-anat./clin. -- 24.44%
Pediatric cardiology -- 20.29%
Pediatrics -- 41.01%
Physical med./rehab. -- 30.10%
Plastic surgery -- 7.12%
Psychiatry -- 24.80%
Public health -- 24.14%
Pulmonary diseases -- 8.84%
Radiation oncology -- 18.79%
Radiology -- 9.99%
Thoracic surgery -- 1.42%
Urological surgery -- 1.71%

(b) Three highest: pediatrics, child psychiatry. and physical med./rehab.
Three lowest: thoracic surgery, urological surgery, and orthopedic surgery

(c) GRAPHICS

(d) (Asks for interpretation)




2-1: Hypothetical Exam Scores

(a) The centers of these distributions vary.

(b) The spreads of these distributions vary.

(c) The shapes of these distributions vary.

(d) This distribution has two distinctive clusters.

(e) This distribution has two outliers- one very low score and one very high.

(f) This distribution reveals granularity in that scores occur only in multiples of five.

2-2: British Rulers' Reigns

(a) 63; Victoria

(b) 0; Edward V; he ruled for some short period less than a year

(c)
0 | 9 0 2 6 5 3 6 7 9 1
1 | 3 9 0 7 3 3 2 3 0 5
2 | 1 0 2 2 4 2 4 5 5
3 | 5 5 5 9 8 3
4 | 4
5 | 6 0 9
6 | 3

(d)
0 | 0 1 2 3 5 6 6 7 9 9
1 | 0 0 2 3 3 3 3 5 7 9
2 | 0 1 2 2 2 4 4 5 5
3 | 3 5 5 5 8 9
4 | 4
5 | 0 6 9
6 | 3

(e) f

(f) 19.5

(g) 9.5

(h) 34

2-3: Pennsylvania College Tuitions

(a) 16

(b) 16; .1019

(c) skewed to the right

(d) There are roughly three clusters or peaks, perhaps corresponding to public institutions and two classes of private ones.

2-4 Students' Measurements

These answers depend on class results.




3-1: Supreme Court Service

(a) GRAPHICS

(b) Many reasonable answers are acceptable.

(c) 76/9 (approx. 8.44)

(d) more: 3; less: 6

(e) 6

(f) more: 4; less: 4

(g) the fifth

(h) n=5: the third
n=7: the fourth
n=9: the fifth
n=11: the sixth
n=13: the seventh

(i) the ((n+1)/2)th

3-2: Faculty Years of Service

(a) 1 3 6 6 7 15 28 31

(b) 12.125

(c) 6

(d) There are an even number of observations, so none falls right in the middle.

(e) 6.5

3-3: Properties of Averages

(a) (Asks for prediction)

(b) Class A -- Mean: 80.55; Median: 81
Class B -- Mean: 69.38; Median: 70
Class C -- Mean: 60.26; Median: 61

(c) - (e) (Asks for prediction)

(f) Class G -- Mean: 49.77; Median: 49
Class H -- Mean: 50.61; Median: 49
Class I -- Mean: 51.20; Median: 53

(g) The mean and median are close together for symmetric distributions. The mean exceeds the median for distributions skewed to the right, and the mean is less than the median for distributions skewed to the left.

(h) - (i) see table below

(j) Justices -- Mean: 8.44; Median: 6
Justices with "big outlier -- Mean: 10.67; Median: 6
Justices with "huge" outlier -- Mean: 30.67; Median: 6

(k) The median is resistant.

(l) no

(m) No, it does not make sense to talk about the mean or median gender. The mode of the genders is sensible, for it is the more frequent gender.

3-4: Readability of Cancer Pamphlets

(a) We do not know the actual reading levels of the 6 patients in the "under 3" category and the 17 patients in the "above 12" category..

(b) 9

(c) 9

(d) The medians are the same.

(e) no

(f) 17/63

3-5: Students' Distances from Home

(a) - (e) These answers depend on class results.

(f) no




4-1: Supreme Court Service ( cont. )

(a) GRAPHICS

(b) mean = 8.44; median = 6

(c) 22

(d) 0 1 3 4; lower quartile = 2

(e) 8 13 19 22; upper quartile = 16

(f) IQR = 14

(g) Original Data -- Deviation from mean -- Squared deviation
22 -- 13.56 -- 188.75
19 -- 10.56 -- 111.42
13 -- 4.56 -- 20.79
8 -- -0.44 -- 0.20
6 -- -2.44 -- 5.95
4 -- -4.44 -- 19.75
3 -- -5.44 -- 29.64
> 1 -- -7.44 -- 55.42
> 0 -- -8.44 -- 71.31
Column sum -- 76 -- .04 -- 503.23

(h) 7.89

(i) GRAPHICS

4-2: Properties of Measures of Spread

(a) (Asks for prediction)

(b) Class D -- Std. dev: 2.837; IQR: 3
Class E -- Std. dev: 7.018; IQR: 9
Class F -- Std. dev: 11.78; IQR: 13.75

(c) - (d) see table below

(e) Justices -- Std. dev: 7.89; IQR: 14
Justices with "big" oultier -- Std. dev: 13.21; IQR: 14
Justices with "huge" outlier -- Std. dev: 72.0; IQR: 14

(f) The interquartile range is resistant. The standard deviation and range are not resistant.

4-3: Placement Exam Scores

(a) yes

(b) (6.362, 14.080)

(c) 146; 0.685

(d) 202; 0.948

(e) 213; 1.000

4-4: SATs and ACTs

(a) 184

(b) 7.4

(c) no

(d) 1.057

(e) 1.423

(f) Kathy

(g) (Asks for interpretation)

(h) Mike: -0.897; Karen: -0.308

(i) Karen

(j) The z-score turns out to be negative when the raw score is less than the mean score.




5-1: Shifting Populations

(a) State -- % change -- Region
Alabama -- 3.6 -- E
Alaska -- 8.9 -- W
Arizona -- 7.4 -- W
Arkansas -- 3.1 -- W
California -- 3.7 -- W
Colorado -- 8.2 -- W
Connecticut -- -0.3 -- E
Delaware -- 5.1 -- E
Florida -- 5.7 -- E
Georgia -- 6.8 -- E
Hawaii -- 5.7 -- W
Idaho -- 9.2 -- W
Illinois -- 2.3 -- E
Indiana -- 3.0 -- E
Iowa -- 9.2 -- W
Kansas -- 2.1 -- W
Kentucky -- 2.8 -- E
Louisiana -- 1.9 -- W
Maine -- 0.9 -- E
Maryland -- 3.8 -- E
Massachusetts -- -0.1 -- E
Michigan -- 2.0 -- E
Minnesota -- 3.3 -- W
Mississippi -- 2.7 -- E
Missouri -- 2.3 -- W
Montana -- 5.1 -- W
Nebraska -- 1.8 -- W
Nevada -- 15.6 -- W
New Hampshire -- 1.4 -- E
New Jersey -- 1.9 -- E
New Mexico -- 6.7 -- W
New York -- 1.1 -- E
North Carolina -- 4.8 -- E
North Dakota -- -0.6 -- W
Ohio -- 2.3 -- E
Oklahoma -- 2.7 -- W
Oregon -- 6.7 -- W
Pennsylvania -- 1.4 -- E
Rhode Island -- -0.3 -- E
South Carolina -- 4.5 -- E
South Dakota -- 2.8 -- W
Tennessee -- 4.5 -- E
Texas -- 6.2 -- W
Utah -- 7.9 -- W
Vermont -- 2.3 -- E
Virginia -- 4.9 -- E
Washington -- 8.0 -- W
West Virginia -- 1.5 -- E
Wisconsin -- 3.0 -- E
Wyoming -- 3.7 -- W

(b) GRAPHICS

(c) East: 2.5; West: 4.4

(d) This answer varies from student to student.

(e) The West tends to have a higher percentage change.

(f) No; there are many possible pairings.

(g) The West.

5-2: Professional Golfers' Winnings

(a) Tour -- Minimum -- Lower quartile -- Median -- Upper quartile -- Maximum
PGA -- 435 -- 809 -- 658 -- 495 -- 1165
LPGA -- 122 -- 301 -- 186 -- 149 -- 863
Senior -- 208 -- 568 -- 340 -- 265 -- 1190

(b) GRAPHICS

(c) no

(d) PGA: no outliers
LPGA: three outliers: 863, 732, 543

(e) GRAPHICS

(f) yes

(g) (Asks for interpretation)

5-3: Ages of Coins

(a) - (d) These answers depend on the class results.




6-1: Cars Fuel Efficiency ( cont. )

(a) GRAPHICS

(b) Heavier cars tended to get worse fuel efficiency.

(c) negatively

(d) yes; there are many such pairs.
GRAPHICS

6-2: Guess the Association

(a) Students who did well on the first test tended to do well on the second, and those who did poorly on the first tended to do poorly on the second.

(b) Negative: Most strong: C; Moderate: D Least Strong: F
Positive: Most Strong: E; Moderate: A; Least Strong: B

(c) (Asks for interpretation)

6-3: Marriage Ages ( cont. )

(a) yes; positive; fairly strong

(b) 2

(c) 16

(d) 6

(e) One can discover that for most of these couples, the husband is older than the wife.

6-4: Fast Food Sandwiches

(a) GRAPHICS

(b) There is a reasonably strong, positive association between a sandwich's serving size and its calories.

(c) Roast beef sandwiches tend to have more calories per serving size than do chicken and turkey sandwiches.

6-5: Space Shuttle O-Ring Failures

(a) GRAPHICS
The scatterplot reveals a weak-to-moderate negative association between temperature and O-ring failures.

(b) The likelihood of O-ring failure appears to be greater at such a low temperature.

(c) GRAPHICS
The association is much weaker in this case.

(d) (Asks for interpretation)





7-1: Properties of Correlation

(a) Negative: Strong: C, -.985; Moderate: D, -.720; Least Strong: F, -.472
Postive: Strong: E, 0.989; Moderate: A, 0.713; Least Strong: B, 0.465

(b) largest: 1; smallest: -1

(c) The data would have to fall exactly on a straight line.

(d) The sign of the correlation (positive or negative) matches the direction of the association.

(e) The stronger the association, the closer the correlation comes to 1. The weaker the association, the closer the association comes to 0.

(f) The data reveal a distinct curvilinear relationship.

(g) 0

(h) Yes, except for the person who scored very high on the first exam and very low on the second.

(i) Yes, except for the person who scored very low on both exams.

(j) H: 0.037; I: 0.705

(k) H: 1.000; I: 0.130; These correlations have changed substantially.

(l) No, because it can be strongly affected by outliers.

(m) There are two distinct clusters, one with students doing poorly on both exams and another with students doing well on both exams.

(n) .954

7-2: Televisions and Life Expectancy

(a) fewest: United States, 1.3; most: Haiti, 234

(b) GRAPHICS

(c) -.804

(d) (Asks for interpretation)

(e) no

(f) (Asks for interpretation)

7-3: Cars' Fuel Efficiency ( cont. )

(a) Model -- Weight -- Weight z-score -- MPG -- MPG z-score -- Product
BMW 3-Series -- 3250 -- 0.07 -- 28 -- 0.07 -- 0.00
BMW 5-Series -- 3675 -- 0.79 -- 23 -- -0.68 -- -0.54
Cadillac Eldorado -- 3840 -- 1.07 -- 19 -- -1.28 -- -1.37
Cadillac Seville -- 3935 -- 1.23 -- 20 -- -1.13 -- -1.39
Ford Aspire -- 2140 -- -1.81 -- 43 -- 2.30 -- -4.17
Ford Crown Victoria -- 4010 -- 1.36 -- 22 -- -0.83 -- -1.13
Ford Escort -- 2565 -- -1.09 -- 34 -- 0.96 -- -1.05
Ford Mustang -- 3450 -- 0.41 -- 22 -- -0.83 -- -0.34
Ford Probe -- 2900 -- -0.52 -- 28 -- 0.07 -- -0.03
Ford Taurus -- 3345 -- 0.23 -- 25 -- -0.38 -- -0.09
Ford Taurus SHO -- 3545 -- 0.57 -- 24 -- -0.53 -- -0.30
Honda Accord -- 3050 -- -0.27 -- 31 -- 0.51 -- -0.14
Honda Civic -- 2540 -- -1.13 -- 34 -- 0.96 -- -1.09
Honda Civic del Sol -- 2410 -- -1.35 -- 36 -- 1.26 -- -1.70
Honda Prelude -- 2865 -- -0.58 -- 30 -- 0.36 -- -0.21
Lincoln Mark VIII -- 3810 -- 1.02 -- 22 -- -0.83 -- -0.83

(b) -.96

(c) The cars with negative wieght z-scores tend to have positive MPG z-scores, and vice versa.

7-4: Guess the Correlation

(a) - (c) The answers vary from student to student.





8-1: Air Fares ( cont. )

(a) - (e) The answers vary from student to student.

8-2: Air Fares ( cont. )

(a) Airfare (y): Mean: 166.9; Std. dev: 59.5
Distance (x): Mean: 713; Std. dev: 413
Correlation: .795

(b) b = 0.117; a = 83.3

(c) airfare = 83.3 + .117 distance

(d) 118.50

(e) 259.30

(f) GRAPHICS

(g) (Asks for prediction)

(h) 188.90

(i) 416.80

(j) Distance -- Predicted Airfare
900 -- 188.90
901 -- 189.02
902 -- 189.13
903 -- 189.25

(k) yes; 0.117; this number is the slope coefficient of the least squares line.

(l) 11.70

(m) 150.88

(n) 27.12

(o) Destination -- Distance -- Airfare -- Fitted -- Residual
Atlanta -- 576 -- 178 -- 150.88 -- 27.12
Boston -- 370 -- 138 -- 126.70 -- 11.30
Chicago -- 612 -- 94 -- 155.10 -- -61.10
Dallas/Fort Worth -- 1216 -- 278 -- 226.00 -- 52.00
Detroit -- 409 -- 158 -- 131.27 -- 26.73
Denver -- 1502 -- 258 -- 259.57 -- -1.56
Miami -- 946 -- 198 -- 194.30 -- 3.70
New Orleans -- 998 -- 188 -- 200.41 -- -12.41
New York -- 189 -- 98 -- 105.45 -- -7.45
Orlando -- 787 -- 179 -- 175.64 -- 3.36
Pittsburgh -- 210 -- 138 -- 107.92 -- 30.08
St. Louis -- 737 -- 98 -- 169.77 -- -71.77

(p) St. Louis; distance: 737 airfare: 98; residual: 71.77; overestimate

(q) greater

(r) below

(s) mean: 0; standard deviation: 36.1

(t) 0.368

(u) 0.632

(v) The sum equals one.

(w) .632

8-3: Students' Measurements ( cont. )

(a) - (e) These answers depend on class results.

(f) no




9-1: Gestation and Longevity

(a) GRAPHICS
Gestation = 21.7 + 13.1 Longevity

(b) For every additional year of longevity, one expects the animal's gestation period to increase by 13.1 days.

(c) .44

(d) GRAPHICS

(e) elephant;
GRAPHICS
No, the elephant does not have the largest residual.

(f) giraffe; longer

(g) Gestation = 9.0 + 13.6 Longevity; 50.1%

(h) no

(i) GRAPHICS
Gestation = 45.0 + 11.1 Longevity; 26.9%

(j) The removal of the elephant affected the graph more.

(k) GRAPHICS
Gestation = 110 + 5.26Longevity
r^2 = .092

9-2: Planetary Measurements ( cont. )

(a) (Asks for interpretation)

(b) 0.910, a very strong, positive correlation which seems to indicate a strong linear relationship.

(c) GRAPHICS
Distance = -1126 + 446 Position

(d) No, the line does not fit that data well. The line does not capture the curved aspect of the relationship.

(e) GRAPHICS
Yes, the residual plot reveals a clear, curved pattern.

9-3: Televisions and Life Expectancy ( cont. )

(a) GRAPHICS
r = -0.804; the relationship does not appear to be linear

(b) GRAPHICS

(c) GRAPHICS
r = -0.855; the relationship appears to be stronger and more linear.

(d) life expectancy = 77.9 - 9.81Log(PerTV)

(e) 85.1%

(f) 55

(g) GRAPHICS
no

(h) The transformed data produces a better fit than the original data.




10-1: Penny Thoughts ( cont. )

(a) TABLE
These answers depend on the class results.

(b) TABLE
These answers depend on the class results.

10-2: Age and Political Ideology

(a) 0.205

(b) 0.388

(c) 0.406

(d) 0.280

(e) 0.473

(f) 0.247

(g) GRAPHICS

(h) (Asks for interpretation)

(i) 0.473

(j) 0.199

(k) 0.097

10-3: Pregnancy, AZT, and HIV

(a) (Asks for prediction)

(b) AZT: 0.079; HIV: 0.250

(c) 3.23

(d) (Asks for interpretation)

10-4: Hypothetical Hospital Recovery Rates

(a) A: 0.8; B: 0.7; B saves the higher percentage.

(b) Are you convinced?

(c) A: 0.983; B: 0.967; A saves the higher percentage.

(d) A: 0.525; B: 0.3; A saves the higher percentage.

(e) Hospital A tends to treat the large majority of patients in "poor" condition. Since these patients are more likely to die than those in "good" condition, A's overall survival rate is lower than B's despite being higher for each type of patient.

(f) Hospital A is preferable regardless of one's condition.

10-5: Hypothetical Employee Retention Predictions

(a) 0.16

(b) 0.16

(c) no

(d) no

(e) GRAPHICS

(f) GRAPHICS




11-1: Elvis Presley and Alf Landon

(a) No; those with more passionate views of Elvis are more likely to pay to voice their opinions.

(b) One flaw is that those with telephones or vehicles in 1936 tended to be the more affluent segment of society. Another flaw is that those who took the time to respond probably tended to be less satisfied with the incumbent.

(c) Elvis: population - all American adults, sample - those radio listeners who chose to pay to call in station
Literary Digest: population - all American adults, sample - those who received the questionnaire and chose to respond

11-2: Sampling U.S. Senators

(a) Gender: categorical (binary)
Party: categorical (binary)
State: categorical
Years of service: measurement

(b) - (e) These answers vary from student to student.

(f) no; no; almost certainly no

(g) No, the random sampling method is not biased because it does not systematically under- or over-represent certain groups.

(h) You would have to use three-digit labels for the representatives, read off three digits at a time from the table, and ignore numbers from 436 to 999 and 000.

(i)

11-3: Sampling U.S. Senators ( cont. )

(a) - (f) Answers vary.

(g) sample size of 10

(h) sample size of 40

(i) the population mean of 12.54

(j) no




12-1: Colors of Reese's Pieces Candies

(a) Answers vary.

(b) statistic, p-hat

(c) parameter, theta

(d) no

(e) yes, answers vary.

(f) no; sampling variability

(g) Answers vary.

(h) no (almost surely)

(i) no (almost surely)

(j) Answers vary.

(k) yes; yes;

(l) more spread out

(m) less spread out

12-2: Simulating Reese's Pieces

(a) Answers vary.

(b) Distribution of sample proportion should be roughly mound shaped, symmetric, and centered in the neighborhood of 0.45.

(c) Answers vary.

(d) yes

(e) Answers vary.

(f) This should be the middle percentage from the table.

(g) no; yes; Since most (about 95%) sample proportions fall within .20 of the population proportion, it is quite likely that a particular sample proportion will fall within .20 of the population proportion.

(h) Answers vary.

(i) less spread out but roughly same shape and center

(j) Answers vary.

(k) There should be a higher percentage of sample proportions falling within .10 of .45 with samples of 75.

(l) larger sample size

(m) - (n) Answers vary.

(o) 95%

(p) Mean: .45; Standard deviation: .0995

(q) Mean: .45; Standard deviation: .0574




13-1: Widget Manufacturing

(a) parameter; theta

(b) p-hat = 4/15

(c) yes

(d) no; sampling variability

(e) - (g) Answers vary.

(h) no (almost surely)

(i) - (j) Answers vary.

(k) no

(l) - (m) Answers vary.

(n) no

(o) no

13-2: Widget Manufacturing ( cont. )

(a) Answers vary.

(b) yes

(c) Answers vary. The evidence of improvement is even stronger here.

13-3: ESP Testing

(a) 0

(b) 4797; 47.97%

(c) According to the simulation, the chance of getting nine or more correct by guessing is around 32.09%, not very surprising.

(d) According to the simulation, the chances of getting twelve or more correct is around 5.05%, moderately surprising.

(e) According to the simulation, the chances of getting fifteen or more correct is around .24%, very surprising.

(f) Only about 3 of every 10,000 subjects would get seventeen or more correct by guessing. This constitutes strong evidence that the person actually possesses some special ability.




14-1: Placement Scores and Reese's Pieces

(a) roughly symmetric, mound shaped

(b) GRAPHICS

(c) GRAPHICS

14-2: Standard Normal Calculations

(a) .7517

(b) .7517

(c) .2483

(d) .0838

(e) .6679

(f) less than .0002

(g) k = 1.28

(h) k = 1.53

14-3: IQ Scores

(a) (Asks for prediction)

(b) -1.00

(c) GRAPHICS

(d) .1587

(e) .9332

(f) .5007

(g) 0.1%

(h) 139.96




15-1: Sampling Reese's Pieces ( cont. )

(a) normal distribution with mean .45 and standard deviation .0574

(b) GRAPHICS

(c) .1922

(d) .9182

(e) They should be close.

15-2: ESP Testing ( cont. )

(a) normal distribution with mean .25 and standard deviation .0791

(b) GRAPHICS

(c) GRAPHICS

(d) .0505

(e) 12 or more correct: 505/10000. They are amazingly close.

(f) Reasonably but not terribly surprising; yes; no

15-3: Effects of Sample Size

(a) normal distribution with mean .45 and standard deviation .0376; same shape and center but less spread out

(b) GRAPHICS

(c) GRAPHICS

(d) .0918; smaller probability

(e) .9922; larger probability




16-1: Penny Spinning

(a) Answers vary.

(b) statistic; p-hat

(c) no

(d) yes

16-2: Critical Values

(a) GRAPHICS

(b) .975

(c) 1.96

(d) z* = 1.44

16-3: Penny Spinning ( cont. )

(a) Answers vary.

(b) no

(c) - (d) Answers vary.

(e) look at z* times the quantity square root of p-hat times one minus p-hat all divided by n

(f) Simple random sampling, large sample size

16-4: Computer Simulations

(a) - (c) Answers vary.

(d) no

(e) 95% of all intervals geberated by this procedure do contain the actual parameter value.

16-5: Effect of Confidence Level

(a) (Asks for prediction)

(b) Answers vary , but the intervals should get gradually wider (with the same center).

(c) Requiring less confidence allows for a narrower interval.

16-6: Effect of Sample Size

(a) (Asks for prediction)

(b) Sample Size -- Sample heads -- Confidence interval -- Half-width -- Width
100 -- 35 -- (.257, .443) -- .093 -- .156
400 -- 140 -- (.303, .397) -- .047 -- .094
800 -- 280 -- (.317, .383) -- .073 -- .066
1600 -- 560 -- (.327, .373) -- .023 -- .046

(c) The intervals get narrower as the sample size gets larger.

(d) twice as big

(e) cuts the half-width in half




17-1: American Moral Decline ( cont. )

(a) (.7294, .7906)

(b) simple random sample; large sample size

(c) .0306

(d) This margin of error comes from the half-width of the 95% confidence interval.

(e)

(f) (Asks for prediction)

(g) .0187; .0419

(h) The margin-of-error decreases as the sample size increases.

(i) Greater, since the sample size of men only would be smaller than that of the complete sample.

(j) Greater, since the sample size of male college graduates would be smaller than that of the complete sample.

17-2: Congressional Term Limits

(a) need 601 people (rounded up from 600.25)

(b) (Asks for prediction)

(c) 9604

(d) (Asks for prediction)

(e) 16,590

(f) not at all

(g) Case -- Sample size -- Confidence level -- C.I. half-width
1 -- Fixed -- Increases -- Increases
2 -- Fixed -- Increases -- Increases
3 -- Increases -- Fixed -- Decreases
4 -- Decreases -- Fixed -- Increases
5 -- Increases -- Increases -- Fixed
6 -- Increases -- Increases -- Fixed

(h) the whole population of American adults

17-3: Female Senators ( cont. )

(a) (.02, .12)

(b) no

(c) horribly biased sampling method

(d) No, because we know the whole population of the 1994 U.S. Senate.

17-4: College Students' Credit

(a) - (b) Answers vary.

(c) It is doubtful that the results would generalize to a larger population.




18-1: ESP Testing ( cont. )

(a) .25

(b) no

(c) yes

(d) yes

(e) Wilma

(f) normal distribution with mean .25 and standard deviation .0433

(g) no

(h) yes

(i) Subject -- Sample number of correct IDs -- Sample proportion of correct IDs -- Approx.probability of doing so well by just guessing -- Your belief that theta > .25 (better than guessing)
Fred -- 28 -- .28 -- .2442 -- none
Barney -- 31 -- .31 -- .0829 -- some
Betty -- 34 -- .34 -- .0188 -- much
Wilma -- 37 -- .37 -- .0028 -- very much

18-2: ESP Testing ( cont. )

(a) Ho: theta = .25
(The subject is just guessing and would get 25% right in long run.)

(b) Ha: theta > .25
(The subject does better then guessing and would get more than 25% right in long run.)

(c) z = 1.39

(d) p-value = .0823

(e) If Barney were just guessing, he'd do this well or better about 8.23% of the time in the long run.

(f) There is some, but not much, evidence to support the claim that Barney does better than just guessing.

(g) yes; yes; no

(h) no; no; no

(i) Subject -- Sample proportion -- Test statistic -- p-value -- Signif. at .10 level? -- Signif. at 0.5? -- Signif. at .01?
Fred -- 0.28 -- 0.69 -- 0.2442 -- no -- no -- no
Barney -- 0.31 -- 1.39 -- 0.0829 -- yes -- no -- no
Betty -- 0.34 -- 2.08 -- 0.0188 -- yes -- yes -- no
Wilma -- 0.37 -- 2.77 -- 0.0028 -- yes -- yes -- yes

18-3: Effect of Sampling Size

(a) parameter, since it pertains to the entire popultation of adult Americans.

(b) Ho: theta = .5; Ha: theta > .5

(c) statistic

(d) sample size

(e) (Asks for prediction)

(f) Sample size -- (One-sided) p-value -- Signif. at .10 level? -- Signif. at .05? -- Signif at .01 level? -- Signif at .001 level?
100 -- .2119 -- no -- no -- no -- no
300 -- .0829 -- yes -- no -- no -- no
500 -- .0368 -- yes -- yes -- no -- no
1000 -- .0059 -- yes -- yes -- yes -- no
2000 -- .0002 -- yes -- yes -- yes -- yes

(g) if the sample size was quite large




19-1: Penny Spinning ( cont. )

(a) theta, the proportion of all penny spins that would land heads

(b) two-sided;
you are looking at "euqally likely" without regard to more or less.

(c) Ho: theta = 0.5; Ha: theta 0.5

(d) z = -1.633; p-value = .1025

(e) z = 1.633; p-value = .1025

(f) no

(g) Ho: theta = 0.5; Ha: theta < 0.5

(h) z = -1.633; p-value = .0512

(i) z = 1.633; p-value = .9488

(j) The sample result is not even in the direction of the alternative hypothesis.

(k) Sample Result -- Alternative hypothesis -- Test statistic -- p-value
65 heads, 85 tails -- theta .5 -- 1.633 -- .1024
85 heads, 65 tails -- theta .5 -- -1.633 -- .1024
65 heads, 85 tails -- theta < .5 -- 1.633 -- .0512
85 heads, 65 tails -- theta < .5 -- -1.633 -- .9488

19-2: American Moral Decline ( cont. )

(a) (.729, .790)

(b) no

(c) - (g) Hypothesized value of theta -- Contained in 95% c.i.? -- Test statistic -- (Two-sided) p-value -- Significant at .05 level?
.50 -- no -- 14.187 -- 0 -- yes
.70 -- no -- 3.543 -- .0004 -- yes
.75 -- yes -- 0.591 -- .5575 -- no
.78 -- yes -- -1.363 -- .1729 -- no
.80 -- yes -- -2.779 -- .0055 -- yes

(h) Whenever the confidence interval includes the value, the test is not significant. Whenever the confidence interval does not include the value, the test is significant.

19-3: Advertising Strategies

(a) z = 1.556; p-value = .0598

(b) z = 1.697; p-value = .0448

(c) z = 3.394 ; p-value = .0003

(d) a and b

(e) b and c

19-4: Tax Return Errors ( cont. )

(a) p-hat = .30626

(b) (Asks for prediction)

(c) Ho: theta = .3 ; Ha: theta .3
z = 3.055; p-value = .0023

(d) (.301, .312)

(e) no

(f) no

19-5: Elvis and Alf ( cont. )

(a) z = 216.887
p-value = 0 (virtually)

(b) (.569, 571)

(c) The sampling procedure was horribly biased in favor of Landon.




20-1: SAT Coaching

(a) The primary difficulty is the lack of a comparison group. Several reasonable explanations can be provided.

(b) explanatory: whether the student had coaching or not;
response: the student's improvement in SAT scores

(c) observational study

20-2: Pet Therapy

(a) explanatory: whether the patient owns a pet or not;
response: whether the patient survives or not

(b) observational study

20-3: Vitamin C and Cold Resistance

(a) expl: whether the subject takes vitamin C or not;
response: whether the subject resists a cold or not

(b) controlled experiment

(c) Many reasonable answers are possible.

20-4: Pregnancy, AZT, and HIV ( cont. )

(a) expl: whether the mother receives AZT or a placebo (binary);
response: whether the baby is HIV positive or not (binary)

(b) One group of women receives AZT and another group recieves a placebo.

(c) Women should be randomly assigned to one of the two groups.

(d) Women should not know to which group they have been assigned.

20-5: Smoking and Lung Cancer

(a) One could randomly assign children to become smokers or nonsmokers and observe whether they develop lung cancer.

(b) case-control

(c) cohort

20-6: SmartFood Popcorn

Many reasonable designs are possible.




21-1: Hypothetical Medical Recovery Rates

(a) N: 0.7; O: 0.5

(b) - (h) Answers vary.

(i) Answers vary, but it should not be very unusual to obtain this sample result by random assignment.

(j) Yes, this sample result should be unusual to achieve by random assignment.

21-2: Pregnancy, AZT, HIV ( cont. )

(a) Ho: theta(AZT) = theta(PLAC); Ha: theta(AZT) = theta(PLAC) (b) p-hat(AZT) = .0793; p-hat(PLAC) = .25

(c) p-hat(c) = .1636

(d) z = -4.54

(e) p-value = 0 (virtually)

(f) virtually 0; yes, extremely unlikely to occur by chance alone

(g) The experimental data provide extremely strong evidence that AZT is more effective than the placebo.

21-3: Hypothetical Medical Recovery Rates ( cont. )

(a) sample sizes

(b) - (c) (Asks for prediction)

(d) "Old" treatment sample size -- "New" treatment sample size -- (One-sided) p-value -- Signigicant at .10? -- Significant at .05? -- Significant at .01?
50 -- 50 -- .1473 -- no -- no -- no
100 -- 100 -- .0691 -- yes -- no -- no
200 -- 200 -- .0180 -- yes -- yes -- no
500 -- 500 -- .0005 -- yes -- yes -- yes

(e) The difference between 60% and 70% is not convincing if very small samples are involved, but it is convincing if large samples are involved.




22-1: Pregnancy, AZT, and HIV ( cont. )

(a) (-.250, -.092)

(b) The proportion of HIV positive babies in the AZT group is less than the proportion of HIV positive babies in the Placebo group by between roughly 9% and 25%.

(c) (-.264, -.077); wider

(d) (.092, .250); this interval is the negative of the interval in (a); the conclusion does not change at all

22-2: Campus Alcohol Habits

(a) p-hat(1982) = .8233; p-hat(1991) = .7884

(b) Ho: theta(1982) = theta(1991); Ha: theta(1982) > theta(1991)

(c) z = 4.431; p-value = 0 (virtually)

(d) (.019, 050); no; the interval supports the conclusion that the proportion of drinkers was higher in 1982 than in 1991

(e) observational study

(f) no

22-3: Berkeley Graduate Admissions ( cont. )

(a) Ho : theta(m) = theta(w)
Ha : theta(m) > theta(w)
z = 9.555; p-value = 0 (virtually); yes, the difference is statistically significant

(b) no; this is an observational study in which Simpson's paradox explains the discrepancy: women tended to apply to the tougher programs to get into





23-1: Parameters vs. Statistics ( cont. )

(a) statistic; x bar

(b) parameter; mu

(c) statistic; s

(d) parameter; sigma

(e) parameter; mu

(f) statistic; x bar

(g) parameter; mu

(h) statistic; s

(i) parameter; mu

(j) parameter; mu

(k) statistic; x bar

23-2: Students' Sleeping Times

(a) Hypothetical sample -- Sample size -- Sample mean -- Sample standard deviation
1 -- 10 -- 7.60 -- .825
2 -- 10 -- 7.60 -- 1.597
3 -- 30 -- 7.60 -- .825
4 -- 30 -- 7.60 -- 1.599

(b) same mean

(c) variability

(d) 1

(e) 1

(f) sample size

(g) 3

(h) 3

(i) 3

(j) 2

(k) sample mean, sample standard deviation, sample size

23-3: Exploring the t -Distribution

(a) GRAPHICS

(b) GRAPHICS

(c) .025

(d) 2.201

(e) greater

(f) 2.069

(g) 2.704

(h) 1.990

23-4: Exploring the t -Distribution ( cont. )

(a) .1

(b) between .025 and .01

(c) between .025 and .01

(d) between .025 and .01

(e) between .005 and .001

(f) less than .005

(g) greater than .2

(h) between .010 and .002

23-5: Students' Sleeping Times ( cont. )

(a) - (c) Hypothetical sample -- Sample size -- Sample mean -- Sample std. dev. -- 95% confidence intervaL -- (One-sided) p-value
1 -- 10 -- 7.6 -- 0.825 -- 7.010, 8.190 -- .080
2 -- 10 -- 7.6 -- 1.597 -- 6.457, 8.743 -- .22
3 -- 30 -- 7.6 -- 0.825 -- 7.292, 7.908 -- .0063
4 -- 30 -- 7.6 -- 1.599 -- 7.003, 8.197 -- .091

(d) Most accurate: 3;
Least accurate: 2;
Most evidence: 3;
Least evidence: 2

23-6: Students' Sleeping Times ( cont. )

(a) - (d) Answers vary.

(e) The confidence interval would be narrower and the p-value smaller.

(f) The confidence interval would be wider and the p-value larger.

(g) The confidence interval would have the same width but be shifted down; the p-value would be smaller.

(h) Answers vary.

(i) No, since the fact that the students got up for an 8:00 am class would affect the amount of sleep they got.




24-1: Students' Travels ( cont. )

(a) - (c) These answers depend on class results

(d) Part of this answer depends on class results. No, this proportion should not be close to 90% because the interval estimates the mean number of states visited in the population and not individuals' values.

(e) cut in half

(f) decrease to one-third its original size

(g) not technically a simple random sample, but probably fairly representative

(h) The answer depends on class results.

24-2: Hypothetical ATM Withdrawals ( cont. )

(a) Sample size -- Sample mean -- Sample std. dev. -- 95% confidence interval
Machine 1: 50 -- 70.0 -- 30.30 -- (61.39, 78.61)
Machine 2: 50 -- 70.0 -- 30.30 -- (61.39, 78.61)
Machine 3: 50 -- 70.0 -- 30.30 -- (61.39, 78.61)

(b) No, the distributions are very different.

24-3: Marriage Ages ( cont. )

(a) Couple # -- Husband -- Wife -- Difference (husband - wife)
1 -- 25 -- 22 -- 3
2 -- 25 -- 32 -- -7
> 3 -- 51 -- 50 -- 1
4 -- 25 -- 25 -- 0
5 -- 38 -- 33 -- 5
6 -- 30 -- 27 -- 3
7 -- 60 -- 45 -- 15
8 -- 54 -- 47 -- 7
9 -- 31 -- 30 -- 1
10 -- 54 -- 44 -- 10
11 -- 23 -- 23 -- 0
12 -- 34 -- 39 -- -5
13 -- 25 -- 24 -- 1
14 -- 23 -- 22 -- 1
15 -- 19 -- 16 -- 3
16 -- 71 -- 73 -- -2
17 -- 26 -- 27 -- -1
18 -- 31 -- 36 -- -5
19 -- 26 -- 24 -- 2
20 -- 62 -- 60 -- 2
21 -- 29 -- 26 -- 3
22 -- 31 -- 23 -- 8
23 -- 29 -- 28 -- 1
24 -- 35 -- 36 -- -1

(b) GRAPHICS

(c) x bar = 1.875; s = 4.812

(d) Ho: mu(D) = 0
Ha: me(D) > 0
t = 1.91
p-value = .034

(e) (.191, 3.559)

(f) (Asks for interpretation)

24-4: Planetary Measurements ( cont. )

(a) (71.22, 2132.78)

(b) No, the interval is senseless since the data do not constitute a sample from a population.




25-1: Hypothetical Commuting Times

(a) no

(b) Yes, route 1 seems to be quicker.

(c) Sample Size -- Sample mean -- Sample standard deviation
Alex 1: 10 -- 28 -- 6
Alex 2: 10 -- 32 -- 6

(d) Sample Size -- Sample mean -- Sample standard deviation
Alex 1: 10 -- 28 -- 6
Alex 2: 10 -- 32 -- 6
Two-sided p-value: .15

(e) no; no; the observed difference in times is not unlikely to have occured by chance.

(f) Barb's centers are farther apart.

(g) Carl's times are less spread out.

(h) Donna has larger samples of times.

(i) Sample Size -- Sample mean -- Sample standard deviation
Barb 1: 10 -- 25 -- 6
Barb 2: 10 -- 35 -- 6
(Barb's two-sided p-value: .0017)
Carl 1: 10 -- 28 -- 3
Carl 2: 10 -- 32 -- 3
(Carl's two-sided p-value: .0080)
Donna 1: 40 -- 28 -- 6
Donna 2: 40 -- 32 -- 6
(Donna's two-sided p-value: .0038)

(j) Barb's sample means are farther apart;
Carl's sample times are less variable (smaller standard deviation);
Donna has larger samples of times.

25-2: Students' Haircut Prices

(a) - (e) These answers depend on class results.

25-3: Trading for Run Production

(a) raw data (at least sample sizes and standard deviation)

(b) Minimum -- Lower quartile -- Median -- Upper quartile -- Maximum
Without McGriff: 0 -- 2 -- 3 -- 5 -- 13
With McGriff: 0 -- 3 -- 5 -- 8 -- 18

(c) Sample size -- Sample mean -- Sample standard deviation
Without McGriff: 94 -- 989 -- 3.074
With McGriff: 68 -- 5.779 -- 3.816

(d) t = -3.19; p-value = .0018

(e) no

(f) no; this is an observational study