Skip to Content
OpenStax Logo
Buy book
  1. Preface
  2. 1 Sampling and Data
    1. Introduction
    2. 1.1 Definitions of Statistics, Probability, and Key Terms
    3. 1.2 Data, Sampling, and Variation in Data and Sampling
    4. 1.3 Levels of Measurement
    5. 1.4 Experimental Design and Ethics
    6. Key Terms
    7. Chapter Review
    8. Homework
    9. References
    10. Solutions
  3. 2 Descriptive Statistics
    1. Introduction
    2. 2.1 Display Data
    3. 2.2 Measures of the Location of the Data
    4. 2.3 Measures of the Center of the Data
    5. 2.4 Sigma Notation and Calculating the Arithmetic Mean
    6. 2.5 Geometric Mean
    7. 2.6 Skewness and the Mean, Median, and Mode
    8. 2.7 Measures of the Spread of the Data
    9. Key Terms
    10. Chapter Review
    11. Formula Review
    12. Practice
    13. Homework
    14. Bringing It Together: Homework
    15. References
    16. Solutions
  4. 3 Probability Topics
    1. Introduction
    2. 3.1 Terminology
    3. 3.2 Independent and Mutually Exclusive Events
    4. 3.3 Two Basic Rules of Probability
    5. 3.4 Contingency Tables and Probability Trees
    6. 3.5 Venn Diagrams
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Bringing It Together: Practice
    12. Homework
    13. Bringing It Together: Homework
    14. References
    15. Solutions
  5. 4 Discrete Random Variables
    1. Introduction
    2. 4.1 Hypergeometric Distribution
    3. 4.2 Binomial Distribution
    4. 4.3 Geometric Distribution
    5. 4.4 Poisson Distribution
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  6. 5 Continuous Random Variables
    1. Introduction
    2. 5.1 Properties of Continuous Probability Density Functions
    3. 5.2 The Uniform Distribution
    4. 5.3 The Exponential Distribution
    5. Key Terms
    6. Chapter Review
    7. Formula Review
    8. Practice
    9. Homework
    10. References
    11. Solutions
  7. 6 The Normal Distribution
    1. Introduction
    2. 6.1 The Standard Normal Distribution
    3. 6.2 Using the Normal Distribution
    4. 6.3 Estimating the Binomial with the Normal Distribution
    5. Key Terms
    6. Chapter Review
    7. Formula Review
    8. Practice
    9. Homework
    10. References
    11. Solutions
  8. 7 The Central Limit Theorem
    1. Introduction
    2. 7.1 The Central Limit Theorem for Sample Means
    3. 7.2 Using the Central Limit Theorem
    4. 7.3 The Central Limit Theorem for Proportions
    5. 7.4 Finite Population Correction Factor
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  9. 8 Confidence Intervals
    1. Introduction
    2. 8.1 A Confidence Interval for a Population Standard Deviation, Known or Large Sample Size
    3. 8.2 A Confidence Interval for a Population Standard Deviation Unknown, Small Sample Case
    4. 8.3 A Confidence Interval for A Population Proportion
    5. 8.4 Calculating the Sample Size n: Continuous and Binary Random Variables
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  10. 9 Hypothesis Testing with One Sample
    1. Introduction
    2. 9.1 Null and Alternative Hypotheses
    3. 9.2 Outcomes and the Type I and Type II Errors
    4. 9.3 Distribution Needed for Hypothesis Testing
    5. 9.4 Full Hypothesis Test Examples
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  11. 10 Hypothesis Testing with Two Samples
    1. Introduction
    2. 10.1 Comparing Two Independent Population Means
    3. 10.2 Cohen's Standards for Small, Medium, and Large Effect Sizes
    4. 10.3 Test for Differences in Means: Assuming Equal Population Variances
    5. 10.4 Comparing Two Independent Population Proportions
    6. 10.5 Two Population Means with Known Standard Deviations
    7. 10.6 Matched or Paired Samples
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Homework
    13. Bringing It Together: Homework
    14. References
    15. Solutions
  12. 11 The Chi-Square Distribution
    1. Introduction
    2. 11.1 Facts About the Chi-Square Distribution
    3. 11.2 Test of a Single Variance
    4. 11.3 Goodness-of-Fit Test
    5. 11.4 Test of Independence
    6. 11.5 Test for Homogeneity
    7. 11.6 Comparison of the Chi-Square Tests
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Homework
    13. Bringing It Together: Homework
    14. References
    15. Solutions
  13. 12 F Distribution and One-Way ANOVA
    1. Introduction
    2. 12.1 Test of Two Variances
    3. 12.2 One-Way ANOVA
    4. 12.3 The F Distribution and the F-Ratio
    5. 12.4 Facts About the F Distribution
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  14. 13 Linear Regression and Correlation
    1. Introduction
    2. 13.1 The Correlation Coefficient r
    3. 13.2 Testing the Significance of the Correlation Coefficient
    4. 13.3 Linear Equations
    5. 13.4 The Regression Equation
    6. 13.5 Interpretation of Regression Coefficients: Elasticity and Logarithmic Transformation
    7. 13.6 Predicting with a Regression Equation
    8. 13.7 How to Use Microsoft Excel® for Regression Analysis
    9. Key Terms
    10. Chapter Review
    11. Practice
    12. Solutions
  15. A | Statistical Tables
  16. B | Mathematical Phrases, Symbols, and Formulas
  17. Index
1.

The populations from which the two samples are drawn are normally distributed.

3.

H0:σ1=σ2H0:σ1=σ2

Ha:σ1<σ2Ha:σ1<σ2

or

H0: σ 1 2  =  σ 2 2 σ 1 2  =  σ 2 2

Ha: σ 1 2 < σ 2 2 σ 1 2 < σ 2 2

5.

4.11

7.

0.7159

9.

No, at the 10% level of significance, we cannot reject the null hypothesis and state that the data do not show that the variation in drive times for the first worker is less than the variation in drive times for the second worker.

11.

2.8674

13.

Cannot accept the null hypothesis. There is enough evidence to say that the variance of the grades for the first student is higher than the variance in the grades for the second student.

15.

0.7414

17.

Each population from which a sample is taken is assumed to be normal.

19.

The populations are assumed to have equal standard deviations (or variances).

21.

4,939.2

23.

2

25.

2,469.6

27.

3.7416

29.

3

31.

13.2

33.

0.825

35.

Because a one-way ANOVA test is always right-tailed, a high F statistic corresponds to a low p-value, so it is likely that we cannot accept the null hypothesis.

37.

The curves approximate the normal distribution.

39.

ten

41.

SS = 237.33; MS = 23.73

43.

0.1614

45.

two

47.

SS = 5,700.4;

MS = 2,850.2

49.

3.6101

51.

Yes, there is enough evidence to show that the scores among the groups are statistically significant at the 10% level.

55.
  1. H 0 σ 1 2 = σ 2 2 H 0 σ 1 2 = σ 2 2
  2. H a :  σ 1 2 σ 1 2 H a :  σ 1 2 σ 1 2
  3. df(num) = 4; df(denom) = 4
  4. F4, 4
  5. 3.00
  6. Check student't solution.
  7. Decision: Cannot reject the null hypothesis; Conclusion: There is insufficient evidence to conclude that the variances are different.
58.

The answers may vary. Sample answer: Home decorating magazines and news magazines have different variances.

60.
  1. H0: = σ 1 2 σ 1 2 = σ 2 2 σ 2 2
  2. Ha: σ 1 2 σ 1 2 σ 1 2 σ 1 2
  3. df(n) = 7, df(d) = 6
  4. F7,6
  5. 0.8117
  6. 0.7825
  7. Check student’s solution.
    1. Alpha: 0.05
    2. Decision: Cannot reject the null hypothesis.
    3. Reason for decision: calculated test statistics is not in the tail of the distribution
    4. Conclusion: There is not sufficient evidence to conclude that the variances are different.
62.

Here is a strip chart of the silver content of the coins:

This graph is a scatterplot which represents the data provided. The horizontal axis is labeled 'Silver content coins' and extends from 5 - 9. The vertical axis is labeled 'Coinage.' The vertical axis is labeled with the categories First, Second, Third, and Fourth.
Figure 12.10

While there are differences in spread, it is not unreasonable to use ANOVA techniques. Here is the completed ANOVA table:

Source of variation Sum of squares (SS) Degrees of freedom (df) Mean square (MS) F
Factor (Between) 37.748 4 – 1 = 3 12.5825 26.272
Error (Within) 11.015 27 – 4 = 23 0.4789
Total 48.763 27 – 1 = 26
Table 12.42

P(F > 26.272) = 0;

Cannot accept the null hypothesis for any alpha. There is sufficient evidence to conclude that the mean silver content among the four coinages are different. From the strip chart, it appears that the first and second coinages had higher silver contents than the third and fourth.

63.

Here is a stripchart of the number of wins for the 14 teams in the AL for the 2012 season.

This graph is a scatterplot which represents the data provided. The horizontal axis is labeled 'Number of wins in 2012 Major League Baseball Season' and extends from 65 - 95. The vertical axis is labeled 'American league division.' The vertical axis is labeled with the categories Central, East, West.
Figure 12.11

While the spread seems similar, there may be some question about the normality of the data, given the wide gaps in the middle near the 0.500 mark of 82 games (teams play 162 games each season in MLB). However, one-way ANOVA is robust.

Here is the ANOVA table for the data:

Source of variation Sum of squares (SS) Degrees of freedom (df) Mean square (MS) F
Factor (Between) 344.16 3 – 1 = 2 172.08
Error (Within) 1,219.55 14 – 3 = 11 110.87 1.5521
Total 1,563.71 14 – 1 = 13
Table 12.43

P(F > 1.5521) = 0.2548
Since the p-value is so large, there is not good evidence against the null hypothesis of equal means. We cannot reject the null hypothesis. Thus, for 2012, there is not any have any good evidence of a significant difference in mean number of wins between the divisions of the American League.

64.

SSbetween = 26
SSwithin = 441
F = 0.2653

67.

df(denom) = 15

69.
  1. H0: µL = µT = µJ
  2. Ha: at least any two of the means are different
  3. df(num) = 2; df(denom) = 12
  4. F distribution
  5. 0.67
  6. 0.5305
  7. Check student’s solution.
  8. Decision:Cannot reject null hypothesis; Conclusion: There is insufficient evidence to conclude that the means are different.
72.
  1. Ha: µc = µn = µh
  2. At least any two of the magazines have different mean lengths.
  3. df(num) = 2, df(denom) = 12
  4. F distribtuion
  5. F = 15.28
  6. p-value = 0.001
  7. Check student’s solution.
    1. Alpha: 0.05
    2. Decision: Cannot accept the null hypothesis.
    3. Reason for decision: p-value < alpha
    4. Conclusion: There is sufficient evidence to conclude that the mean lengths of the magazines are different.
74.
  1. H0: μo = μh = μf
  2. At least two of the means are different.
  3. df(n) = 2, df(d) = 13
  4. F2,13
  5. 0.64
  6. 0.5437
  7. Check student’s solution.
    1. Alpha: 0.05
    2. Decision: Cannot reject the null hypothesis.
    3. Reason for decision: p-value > alpha
    4. Conclusion: The mean scores of different class delivery are not different.
76.
  1. H0: μp = μm = μh
  2. At least any two of the means are different.
  3. df(n) = 2, df(d) = 12
  4. F2,12
  5. 3.13
  6. 0.0807
  7. Check student’s solution.
    1. Alpha: 0.05
    2. Decision: Cannot reject the null hypothesis.
    3. Reason for decision: p-value > alpha
    4. Conclusion: There is not sufficient evidence to conclude that the mean numbers of daily visitors are different.
78.

The data appear normally distributed from the chart and of similar spread. There do not appear to be any serious outliers, so we may proceed with our ANOVA calculations, to see if we have good evidence of a difference between the three groups.

H0:µ1=µ2=µ3H0:µ1=µ2=µ3;

Ha:µi;Ha:µi; some ijij

Define μ1, μ2, μ3, as the population mean number of eggs laid by the three groups of fruit flies.

F statistic = 8.6657;

p-value = 0.0004

This graph shows a nonsymmetrical F distribution curve. This curve does not have a peak, but slopes downward from a maximum value at (0, 1.0) and approaches the horizontal axis at the right edge of the graph.
Figure 12.12

Decision: Since the p-value is less than the level of significance of 0.01, we reject the null hypothesis.

Conclusion: We have good evidence that the average number of eggs laid during the first 14 days of life for these three strains of fruitflies are different.

Interestingly, if you perform a two sample t-test to compare the RS and NS groups they are significantly different (p = 0.0013). Similarly, SS and NS are significantly different (p = 0.0006). However, the two selected groups, RS and SS are not significantly different (p = 0.5176). Thus we appear to have good evidence that selection either for resistance or for susceptibility involves a reduced rate of egg production (for these specific strains) as compared to flies that were not selected for resistance or susceptibility to DDT. Here, genetic selection has apparently involved a loss of fecundity.

Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/introductory-business-statistics/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/introductory-business-statistics/pages/1-introduction
Citation information

© Nov 29, 2017 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.