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Summary

Many times researchers are interested in comparing two parameters such as two means, two proportions, or two variances. When both population standard deviations are known, the z test can be used to compare two means. If both population standard deviations are not known, the sample standard deviations can be used, but the t test is used to compare two means. A z test can be used to compare two proportions. Finally, two variances can be compared using an F test.

Important Formulas

Formula for the z test for comparing two means from independent populations;

σ 1 and σ2 are known:

Formula for the confidence interval for difference of two means when

σ 1 and σ2 are known:

Formula for the t test for comparing two means (independent samples, variances not equal),

σ 1 and σ2 are unknown:

and d.f. = the smaller of n1 − 1 or n2 − 1

Formula for the confidence interval for the difference of two means (independent samples, variances unequal), σ1 and σ2 are unknown:

and d.f. = smaller of n1 − 1 or n2 − 2

Formula for the t test for comparing two means from dependent samples:

where is the mean of the differences

and sD is the standard deviation of the differences

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Formula for confidence interval for the mean of the difference for dependent samples:

and d.f. = n − 1.

Formula for the z test for comparing two proportions:

where

Formula for confidence interval for the difference of two proportions:

Formula for the F test for comparing two variances:

problem-questionReview Exercises

For each exercise, perform these steps. Assume that all variables are normally or approximately normally distributed.

  1. State the hypotheses and identify the claim.

  2. Find the critical value(s).

  3. Compute the test value.

  4. Make the decision.

  5. Summarize the results.

Use the traditional method of hypothesis testing unless otherwise specified.

  1. Driving for Pleasure Two groups of drivers are surveyed to see how many miles per week they drive for pleasure trips. The data are shown. At α = 0.01, can it be concluded that single drivers do more driving for pleasure trips on average than married drivers? Assume σ1 = 16.7 and σ2 = 16.1.

    Single drivers

    Married drivers

    106

    110

    115

    121

    132

    97

    104

    138

    102

    115

    119

    97

    118

    122

    135

    133

    120

    119

    136

    96

    110

    117

    116

    138

    142

    139

    108

    117

    145

    114

    115

    114

    103

    98

    99

    140

    136

    113

    113

    150

    108

    117

    152

    147

    117

    101

    114

    116

    113

    135

    154

    86

    115

    116

    104

    115

    109

    147

    106

    88

    107

    133

    138

    142

    140

    113

    119

    99

    108

    105

  2. Average Earnings of College Graduates The average yearly earnings of male college graduates (with at least a bachelor's degree) are $58,500 for men aged 25 to 34. The average yearly earnings of female college graduates with the same qualifications are $49,339. Based on the results below, can it be concluded that there is a difference in mean earnings between male and female college graduates? Use the 0.01 level of significance.

     

    Male

    Female

    Sample mean

    $59,235

    $52,487

    Population standard deviation

    8,945

    10,125

    Sample size

    40

    35

    Source: New York Times Almanac.
  3. Communication Times According to the Bureau of Labor Statistics’ American Time Use Survey (ATUS), married persons spend an average of 8 minutes per day on phone calls, mail, and e-mail, while single persons spend an average of 14 minutes per day on these same tasks. Based on the following information, is there sufficient evidence to conclude that single persons spend, on average, a greater time each day communicating? Use the 0.05 level of significance.

     

    Single

    Married

    Sample size

    26

    20

    Sample mean

    16.7 minutes

    12.5 minutes

    Sample variance

    8.41

    10.24

    Source: Time magazine.
  4. Average Temperatures The average temperatures for a 25-day period for Birmingham, Alabama, and Chicago, Illinois, are shown. Based on the samples, at α = 0.10, can it be concluded that it is warmer in Birmingham?

     

    Birmingham

    Chicago

    78

    82

    68

    67

    68

    70

    74

    73

    60

    77

    75

    73

    75

    64

    68

    71

    72

    71

    74

    76

    62

    73

    77

    78

    79

    71

    80

    65

    70

    83

    74

    72

    73

    78

    68

    67

    76

    75

    62

    65

    73

    79

    82

    71

    66

    66

    65

    77

    66

    64

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  6. Teachers’ Salaries A sample of 15 teachers from Rhode Island has an average salary of $35,270, with a standard deviation of $3256. A sample of 30 teachers from New York has an average salary of $29,512, with a standard deviation of $1432. Is there a significant difference in teachers’ salaries between the two states? Use α = 0.02. Find the 98% confidence interval for the difference of the two means.

  7. Soft Drinks in School The data show the amounts (in thousands of dollars) of the contracts for soft drinks in local school districts. At α = 0.10 can it be concluded that there is a difference in the averages? Use the P-value method. Give a reason why the result would be of concern to a cafeteria manager.

    Pepsi

    Coca-Cola

    46

    120

    80

    500

    100

    59

    420

    285

    57

    Source: Local school districts.
  8. High and Low Temperatures March is a month of variable weather in the Northeast. The chart below records the actual high and low temperatures for a selection of days in March from the weather report for Pittsburgh, Pennsylvania. At the 0.01 level of significance, is there sufficient evidence to conclude that there is more than a 10° difference between average highs and lows?

    Maximum

    44

    46

    46

    36

    34

    36

    57

    62

    73

    53

    Minimum

    27

    34

    24

    19

    19

    26

    33

    57

    46

    26

  9. Automobile Part Production In an effort to increase production of an automobile part, the factory manager decides to play music in the manufacturing area. Eight workers are selected, and the number of items each produced for a specific day is recorded. After one week of music, the same workers are monitored again. The data are given in the table. At α = 0.05, can the manager conclude that the music has increased production?

    Worker

    1

    2

    3

    4

    5

    6

    7

    8

    Before

    6

    8

    10

    9

    5

    12

    9

    7

    After

    10

    12

    9

    12

    8

    13

    8

    10

  10. Foggy Days St. Petersburg, Russia, has 207 foggy days out of 365 days while Stockholm, Sweden, has 166 foggy days out of 365. At α = 0.02, can it be concluded that the proportions of foggy days for the two cities are different? Find the 98% confidence interval for the difference of the two proportions.

    Source: Jack Williams, USA TODAY.
  11. Adopted Pets According to the 2005–2006 National Pet Owners Survey, only 16% of pet dogs were adopted from an animal shelter and 15% of pet cats were adopted. To test this difference in proportions of adopted pets, a survey was taken in a local region. Is there sufficient evidence to conclude that there is a difference in proportions? Use α = 0.05.

     

    Dogs

    Cats

    Number

    180

    200

    Adopted

    36

    30

    Source: www.hsus.org
  12. Noise Levels in Hospitals In the hospital study cited in Exercise 19 in Exercise set 8-1, the standard deviation of the noise levels of the 11 intensive care units was 4.1 dBA, and the standard deviation of the noise levels of 24 nonmedical care areas, such as kitchens and machine rooms, was 13.2 dBA. At α = 0.10, is there a significant difference between the standard deviations of these two areas?

    Source: M. Bayo, A. Garcia, and A. Garcia, “Noise Levels in an Urban Hospital and Workers’ Subjective Responses,” Archives of Environmental Health.
  13. Heights of World Famous Cathedrals The heights (in feet) for a random sample of world famous cathedrals are listed below. In addition, the heights for a sample of the tallest buildings in the world are listed. Is there sufficient evidence at α = 0.05 to conclude a difference in the variances in height between the two groups?

    Cathedrals

    72

    114 157

    56

    83

    108

    90 151

     

    Tallest buildings

    452

    442 415

    391

    355

    344

    310 302

    209

Statistics TodayTo Vaccinate or Not to Vaccinate? Small or Large?–Revisited

Using a z test to compare two proportions, the researchers found that the proportion of residents in smaller nursing homes who were vaccinated (80.8%) was statistically greater than that of residents in large nursing homes who were vaccinated (68.7%). Using statistical methods presented in later chapters, they also found that the larger size of the nursing home and the lower frequency of vaccination were significant predictions of influenza outbreaks in nursing homes.

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Data Analysis

The Data Bank is found in Appendix D , or on the World Wide Web by following links from www.mhhe.com/math/stat/bluman/

  1. From the Data Bank, select a variable and compare the mean of the variable for a random sample of at least 30 men with the mean of the variable for the random sample of at least 30 women. Use a z test.

  2. Repeat the experiment in Exercise 1, using a different variable and two samples of size 15. Compare the means by using a t test.

  3. Compare the proportion of men who are smokers with the proportion of women who are smokers. Use the data in the Data Bank. Choose random samples of size 30 or more. Use the z test for proportions.

  4. Select two samples of 20 values from the data in Data Set IV in Appendix D. Test the hypothesis that the mean heights of the buildings are equal.

  5. Using the same data obtained in Exercise 4, test the hypothesis that the variances are equal.

Chapter Quiz

Determine whether each statement is true or false. If the statement is false, explain why.

  1. When you are testing the difference between two means for small samples, it is not important to distinguish whether the samples are independent of each other.

  2. If the same diet is given to two groups of randomly selected individuals, the samples are considered to be dependent.

  3. When computing the F test value, you always place the larger variance in the numerator of the fraction.

  4. Tests for variances are always two-tailed.

Select the best answer.

  1. To test the equality of two variances, you would use a(n) ________test.

    1. z

    2. t

    3. Chi-square

    • F

  2. To test the equality of two proportions, you would use a(n) _________test.

    • z

    • b.

      t

    • c.

      Chi-square

    • d.

      F

  3. The mean value of the F is approximately equal to

    1. 0

    2. 0.5

    • 1

    • It cannot be determined.

  4. What test can be used to test the difference between two sample means when the population variances are known?

    • z

    • b.

      t

    • c.

      Chi-square

    • d.

      F

Complete these statements with the best answer.

  1. If you hypothesize that there is no difference between means, this is represented as H0:__________

  2. When you are testing the difference between two means, the ___________ test is used when the population variances are not known.

  3. When the t test is used for testing the equality of two means, the populations must be _____________.

  4. The values of F cannot be _________.

  5. The formula for the F test for variances is _________.

For each of these problems, perform the following steps.

  1. State the hypotheses and identify the claim.

  2. Find the critical value(s).

  3. Compute the test value.

  4. Make the decision.

  5. Summarize the results.

Use the traditional method of hypothesis testing unless otherwise specified.

  1. Cholesterol Levels A researcher wishes to see if there is a difference in the cholesterol levels of two groups of men. A random sample of 30 men between the ages of 25 and 40 is selected and tested. The average level is 223. A second sample of 25 men between the ages of 41 and 56 is selected and tested. The average of this group is 229. The population standard deviation for both groups is 6. At α = 0.01, is there a difference in the cholesterol levels between the two groups? Find the 99% confidence interval for the difference of the two means.

  2. Apartment Rental Fees The data shown are the rental fees (in dollars) for two random samples of apartments in a large city. At α = 0.10, can it be concluded that the average rental fee for apartments in the East is greater than the average rental fee in the West? Assume σ1 = 119 and σ2 = 103.

     

    EastWest

    495

    390

    540

    445

    420

    525

    400

    310

    375

    750

    410

    550

    499

    500

    550

    390

    795

    554

    450

    370

    389

    350

    450

    530

    350

    385

    395

    425

    500

    550

    375

    690

    325

    350

    799

    380

    400

    450

    365

    425

    475

    295

    350

    485

    625

    375

    360

    425

    400

    475

    275

    450

    440

    425

    675

    400

    475

    430

    410

    450

    625

    390

    485

    550

    650

    425

    450

    620

    500

    400

    685

    385

    450

    550

    425

    295

    350

    300

    360

    400

    Source: Pittsburgh Post-Gazette.
  3. Page 527
  4. Prices of Low-Calorie Foods The average price of a sample of 12 bottles of diet salad dressing taken from different stores is $1.43. The standard deviation is $0.09. The average price of a sample of 16 low-calorie frozen desserts is $1.03. The standard deviation is $0.10. At α = 0.01, is there a significant difference in price? Find the 99% confidence interval of the difference in the means.

  5. Jet Ski Accidents The data shown represent the number of accidents people had when using jet skis and other types of wet bikes. At α = 0.05, can it be concluded that the average number of accidents per year has increased from one period to the next?

    1987–19911992–1996

      376

      650

    844

    1650

    2236

    3002

    1162

    1513

     

    4028

    4010

     

    Source: USA TODAY.
  6. Salaries of Chemists A sample of 12 chemists from Washington state shows an average salary of $39,420 with a standard deviation of $1659, while a sample of 26 chemists from New Mexico has an average salary of $30,215 with a standard deviation of $4116. Is there a significant difference between the two states in chemists’ salaries at α = 0.02? Find the 98% confidence interval of the difference in the means.

  7. Family Incomes The average income of 15 families who reside in a large metropolitan East Coast city is $62,456. The standard deviation is $9652. The average income of 11 families who reside in a rural area of the Midwest is $60,213, with a standard deviation of $2009. At α = 0.05, can it be concluded that the families who live in the cities have a higher income than those who live in the rural areas? Use the P-value method.

  8. Mathematical Skills In an effort to improve the mathematical skills of 10 students, a teacher provides a weekly 1-hour tutoring session for the students. A pretest is given before the sessions, and a posttest is given after. The results are shown here. At α = 0.01, can it be concluded that the sessions help to improve the students’ mathematical skills?

    Student

    1

    2

    3

    4

    5

    6

    7

    8

    9

    10

    Pretest

    82

    76

    91

    62

    81

    67

    71

    69

    80

    85

    Posttest

    88

    80

    98

    80

    80

    73

    74

    78

    85

    93

  9. Egg Production To increase egg production, a farmer decided to increase the amount of time the lights in his hen house were on. Ten hens were selected, and the number of eggs each produced was recorded. After one week of lengthened light time, the same hens were monitored again. The data are given here. At α = 3 0.05, can it be concluded that the increased light time increased egg production?

    Hen

    1

    2

    3

    4

    5

    6

    7

    8

    9

    10

    Before

    4

    3

    8

    7

    6

    4

    9

    7

    6

    5

    After

    6

    5

    9

    7

    4

    5

    10

    6

    9

    6

  10. Factory Worker Literacy Rates In a sample of 80 workers from a factory in city A, it was found that 5% were unable to read, while in a sample of 50 workers in city B, 8% were unable to read. Can it be concluded that there is a difference in the proportions of nonreaders in the two cities? Use α = 0.10. Find the 90% confidence interval for the difference of the two proportions.

  11. Male Head of Household A recent survey of 200 households showed that 8 had a single male as the head of household. Forty years ago, a survey of 200 households showed that 6 had a single male as the head of household. At α = 0.05, can it be concluded that the proportion has changed? Find the 95% confidence interval of the difference of the two proportions. Does the confidence interval contain 0? Why is this important to know?

    Source: Based on data from the U.S. Census Bureau.
  12. Money Spent on Road Repair A politician wishes to compare the variances of the amount of money spent for road repair in two different counties. The data are given here. At α = 0.05, is there a significant difference in the variances of the amounts spent in the two counties? Use the P-value method.

    County ACounty B

  13. Heights of Basketball Players A researcher wants to compare the variances of the heights (in inches) of four-year college basketball players with those of players in junior colleges. A sample of 30 players from each type of school is selected, and the variances of the heights for each type are 2.43 and 3.15, respectively. At α = 0.10, is there a significant difference between the variances of the heights in the two types of schools?

Page 528
Critical Thinking Challenges
  1. The study cited in the article entitled “Only the Timid Die Young” stated that “Timid rats were 60% more likely to die at any given time than were their outgoing brothers.” Based on the results, answer the following questions.

    1. Why were rats used in the study?

    2. What are the variables in the study?

    3. Why were infants included in the article?

    4. What is wrong with extrapolating the results to humans?

    5. Suggest some ways humans might be used in a study of this type.

    Reprinted with permission from Psychology Today, Copyright © 2004, Sussex Publishers, Inc.
  2. Based on the study presented in the article entitled “Sleeping Brain, Not at Rest,” answer these questions.

    1. What were the variables used in the study?

    2. How were they measured?

    3. Suggest a statistical test that might have been used to arrive at the conclusion.

    4. Based on the results, what would you suggest for students preparing for an exam?

 
Reprinted with permission from Psychology Today, Copyright © 2004, Sussex Publishers, Inc.
Page 529
Data Projects

Use a significance level of 0.05 for all tests below.

  1. Business and Finance Use the data collected in data project 1 of Chapter 2 to complete this problem. Test the claim that the mean earnings per share for Dow Jones stocks are greater than for NASDAQ stocks.

  2. Sports and Leisure Use the data collected in data project 2 of Chapter 7 regarding home runs for this problem. Test the claim that the mean number of home runs hit by the American League sluggers is the same as the mean for the National League.

  3. Technology Use the cell phone data collected for data project 2 in Chapter 8 to complete this problem. Test the claim that the mean length for outgoing calls is the same as that for incoming calls. Test the claim that the standard deviation for outgoing calls is more than that for incoming calls.

  4. Health and Wellness Use the data regarding BMI that were collected in data project 6 of Chapter 7 to complete this problem. Test the claim that the mean BMI for males is the same as that for females. Test the claim that the standard deviation for males is the same as that for females.

  5. Politics and Economics Use data from the last Presidential election to categorize the 50 states as “red” or “blue” based on who was supported for President in that state, the Democratic or Republican candidate. Use the data collected in data project 5 of Chapter 2 regarding income. Test the claim that the mean incomes for red states and blue states are equal.

  6. Your Class Use the data collected in data project 6 of Chapter 2 regarding heart rates. Test the claim that the heart rates after exercise are more variable than the heart rates before exercise.

Answers to Applying the Concepts
Home Runs
  1. The population is all home runs hit by major league baseball players.

  2. A cluster sample was used.

  3. Answers will vary. While this sample is not representative of all major league baseball players per se, it does allow us to compare the leaders in each league.

  4. H 0: μ1 = μ2 and H1: μ1μ2

  5. Answers will vary. Possible answers include the 0.05 and 0.01 significance levels.

  6. We will use the z test for the difference in means.

  7. Our test statistic is , and our P-value is 0.3124.

  8. We fail to reject the null hypothesis.

  9. Page 530
  10. There is not enough evidence to conclude that there is a difference in the number of home runs hit by National League versus American League baseball players.

  11. Answers will vary. One possible answer is that since we do not have a random sample of data from each league, we cannot answer the original question asked.

  12. Answers will vary. One possible answer is that we could get a random sample of data from each league from a recent season.

Too Long on the Telephone
  1. These samples are independent.

  2. There were 56 + 2 = 58 people in the study.

  3. We compare the P-value of 0.06317 to the significance level to check if the null hypothesis should be rejected.

  4. The P-value of 0.06317 also gives the probability of a type I error.

  5. The F value of 3.07849 is the result of dividing the two sample variances.

  6. Since two critical values are shown, we know that a two-tailed test was done.

  7. Since the P-value of 0.06317 is greater than the significance value of 0.05, we fail to reject the null hypothesis and find that we do not have enough evidence to conclude that there is a difference in the lengths of telephone calls made by employees in the two divisions of the company.

  8. If the significance level had been 0.10, we would have rejected the null hypothesis, since the P-value would have been less than the significance level.

Air Quality
  1. The purpose of the study is to determine if the air quality in the United States has changed over the past 2 years.

  2. These are dependent samples, since we have two readings from each of 10 metropolitan areas.

  3. The hypotheses we will test are H0: μD = 0 and H1: μD ≠ 0.

  4. We will use the 0.05 significance level and critical values of t = ±2.262.

  5. We will use the t test for dependent samples.

  6. There are 10 − 1 = 9 degrees of freedom.

  7. Our test statistic is . We fail to reject the null hypothesis and find that there is not enough evidence to conclude that the air quality in the United States has changed over the past 2 years.

  8. No, we could not use an independent means test since we have two readings from each metropolitan area.

  9. Answers will vary. One possible answer is that there are other measures of air quality that we could have examined to answer the question.

Smoking and Education
  1. Our hypotheses are H0: p1 = p2 and H1: p1p2.

  2. At the 0.05 significance level, our critical values are z = ±1.96.

  3. We will use the z test for the difference between proportions.

  4. To complete the statistical test, we would need the sample sizes.

  5. Knowing the sample sizes were 1000, we can now complete the test.

  6. Our test statistic is , and our P-value is very close to zero. We reject the null hypothesis and find that there is enough evidence to conclude that there is a difference in the proportions of high school graduates and college graduates who smoke.

Variability and Automatic Transmissions
  1. The null hypothesis is that the variances are the same:

  2. We will use an F test.

  3. The value of the test statistic is and the P-value is 0.0008. There is a significant difference in the variability of the prices between the two countries.

  4. Small sample sizes are highly impacted by outliers.

  5. The degrees of freedom for the numerator and denominator are both 5.

  6. Yes, two sets of data can center on the same mean but have very different standard deviations.

Page 531
Hypothesis-Testing Summary 1
  1. Comparison of a sample mean with a specific population mean.

    Example: H0: μ = 100

    1. Use the z test when sigma; is known:

    2. Use the t test when sigma; is unknown:

  2. Comparison of a sample variance or standard deviation with a specific population variance or standard deviation.

    Example: H0: σ2 = 225

    Use the chi-square test:

  3. Comparison of two sample means.

    Example: H0: μ1 = μ2

    1. Use the z test when the population variances are known:

    2. Use the t test for independent samples when the population variances are unknown and assume the sample variances are unequal:

      with d.f. = the smaller of n1 – 1or n2 – 1.

    3. Use the t test for means for dependent samples:

      Example: H0: μD = 0

      where n = number of pairs.

  4. Comparison of a sample proportion with a specific population proportion.

    Example: H0: p = 0.32

    Use the z test:

  5. Comparison of two sample proportions.

    Example: H0: p1 = p2

    Use the z test:

    where

  6. Comparison of two sample variances or standard deviations.

    Example:

    Use the F test:

    where