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Introductory Statistics

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Table of contents
  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 Frequency, Frequency Tables, and Levels of Measurement
    5. 1.4 Experimental Design and Ethics
    6. 1.5 Data Collection Experiment
    7. 1.6 Sampling Experiment
    8. Key Terms
    9. Chapter Review
    10. Practice
    11. Homework
    12. Bringing It Together: Homework
    13. References
    14. Solutions
  3. 2 Descriptive Statistics
    1. Introduction
    2. 2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs
    3. 2.2 Histograms, Frequency Polygons, and Time Series Graphs
    4. 2.3 Measures of the Location of the Data
    5. 2.4 Box Plots
    6. 2.5 Measures of the Center of the Data
    7. 2.6 Skewness and the Mean, Median, and Mode
    8. 2.7 Measures of the Spread of the Data
    9. 2.8 Descriptive Statistics
    10. Key Terms
    11. Chapter Review
    12. Formula Review
    13. Practice
    14. Homework
    15. Bringing It Together: Homework
    16. References
    17. 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
    6. 3.5 Tree and Venn Diagrams
    7. 3.6 Probability Topics
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Bringing It Together: Practice
    13. Homework
    14. Bringing It Together: Homework
    15. References
    16. Solutions
  5. 4 Discrete Random Variables
    1. Introduction
    2. 4.1 Probability Distribution Function (PDF) for a Discrete Random Variable
    3. 4.2 Mean or Expected Value and Standard Deviation
    4. 4.3 Binomial Distribution
    5. 4.4 Geometric Distribution
    6. 4.5 Hypergeometric Distribution
    7. 4.6 Poisson Distribution
    8. 4.7 Discrete Distribution (Playing Card Experiment)
    9. 4.8 Discrete Distribution (Lucky Dice Experiment)
    10. Key Terms
    11. Chapter Review
    12. Formula Review
    13. Practice
    14. Homework
    15. References
    16. Solutions
  6. 5 Continuous Random Variables
    1. Introduction
    2. 5.1 Continuous Probability Functions
    3. 5.2 The Uniform Distribution
    4. 5.3 The Exponential Distribution
    5. 5.4 Continuous Distribution
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. 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 Normal Distribution (Lap Times)
    5. 6.4 Normal Distribution (Pinkie Length)
    6. Key Terms
    7. Chapter Review
    8. Formula Review
    9. Practice
    10. Homework
    11. References
    12. Solutions
  8. 7 The Central Limit Theorem
    1. Introduction
    2. 7.1 The Central Limit Theorem for Sample Means (Averages)
    3. 7.2 The Central Limit Theorem for Sums
    4. 7.3 Using the Central Limit Theorem
    5. 7.4 Central Limit Theorem (Pocket Change)
    6. 7.5 Central Limit Theorem (Cookie Recipes)
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Homework
    12. References
    13. Solutions
  9. 8 Confidence Intervals
    1. Introduction
    2. 8.1 A Single Population Mean using the Normal Distribution
    3. 8.2 A Single Population Mean using the Student t Distribution
    4. 8.3 A Population Proportion
    5. 8.4 Confidence Interval (Home Costs)
    6. 8.5 Confidence Interval (Place of Birth)
    7. 8.6 Confidence Interval (Women's Heights)
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Homework
    13. References
    14. 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 Rare Events, the Sample, Decision and Conclusion
    6. 9.5 Additional Information and Full Hypothesis Test Examples
    7. 9.6 Hypothesis Testing of a Single Mean and Single Proportion
    8. Key Terms
    9. Chapter Review
    10. Formula Review
    11. Practice
    12. Homework
    13. References
    14. Solutions
  11. 10 Hypothesis Testing with Two Samples
    1. Introduction
    2. 10.1 Two Population Means with Unknown Standard Deviations
    3. 10.2 Two Population Means with Known Standard Deviations
    4. 10.3 Comparing Two Independent Population Proportions
    5. 10.4 Matched or Paired Samples
    6. 10.5 Hypothesis Testing for Two Means and Two Proportions
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Homework
    12. Bringing It Together: Homework
    13. References
    14. Solutions
  12. 11 The Chi-Square Distribution
    1. Introduction
    2. 11.1 Facts About the Chi-Square Distribution
    3. 11.2 Goodness-of-Fit Test
    4. 11.3 Test of Independence
    5. 11.4 Test for Homogeneity
    6. 11.5 Comparison of the Chi-Square Tests
    7. 11.6 Test of a Single Variance
    8. 11.7 Lab 1: Chi-Square Goodness-of-Fit
    9. 11.8 Lab 2: Chi-Square Test of Independence
    10. Key Terms
    11. Chapter Review
    12. Formula Review
    13. Practice
    14. Homework
    15. Bringing It Together: Homework
    16. References
    17. Solutions
  13. 12 Linear Regression and Correlation
    1. Introduction
    2. 12.1 Linear Equations
    3. 12.2 Scatter Plots
    4. 12.3 The Regression Equation
    5. 12.4 Testing the Significance of the Correlation Coefficient
    6. 12.5 Prediction
    7. 12.6 Outliers
    8. 12.7 Regression (Distance from School)
    9. 12.8 Regression (Textbook Cost)
    10. 12.9 Regression (Fuel Efficiency)
    11. Key Terms
    12. Chapter Review
    13. Formula Review
    14. Practice
    15. Homework
    16. Bringing It Together: Homework
    17. References
    18. Solutions
  14. 13 F Distribution and One-Way ANOVA
    1. Introduction
    2. 13.1 One-Way ANOVA
    3. 13.2 The F Distribution and the F-Ratio
    4. 13.3 Facts About the F Distribution
    5. 13.4 Test of Two Variances
    6. 13.5 Lab: One-Way ANOVA
    7. Key Terms
    8. Chapter Review
    9. Formula Review
    10. Practice
    11. Homework
    12. References
    13. Solutions
  15. A | Review Exercises (Ch 3-13)
  16. B | Practice Tests (1-4) and Final Exams
  17. C | Data Sets
  18. D | Group and Partner Projects
  19. E | Solution Sheets
  20. F | Mathematical Phrases, Symbols, and Formulas
  21. G | Notes for the TI-83, 83+, 84, 84+ Calculators
  22. H | Tables
  23. Index

12.1 Linear Equations

Use the following information to answer the next three exercises. A vacation resort rents SCUBA equipment to certified divers. The resort charges an up-front fee of $25 and another fee of $12.50 an hour.

1.

What are the dependent and independent variables?

2.

Find the equation that expresses the total fee in terms of the number of hours the equipment is rented.

3.

Graph the equation from Exercise 12.2.


Use the following information to answer the next two exercises. A credit card company charges $10 when a payment is late, and $5 a day each day the payment remains unpaid.

4.

Find the equation that expresses the total fee in terms of the number of days the payment is late.

5.

Graph the equation from Exercise 12.4.

6.

Is the equation y = 10 + 5x – 3x2 linear? Why or why not?

7.

Which of the following equations are linear?

a. y = 6x + 8

b. y + 7 = 3x

c. y – x = 8x2

d. 4y = 8

8.

Does the graph show a linear equation? Why or why not?

This is a graph of an equation. The x-axis is labeled in intervals of 1 from -5 to 5; the y-axis is labeled in intervals of 1 from 0 - 8. The equation's graph is a parabola, a u-shaped curve that has a minimum value at (0, 0).
Figure 12.25

Table 12.12 contains real data for the first two decades of flu reporting.

Year # flu cases diagnosed # flu deaths
Pre-1981 91 29
1981 319 121
1982 1,170 453
1983 3,076 1,482
1984 6,240 3,466
1985 11,776 6,878
1986 19,032 11,987
1987 28,564 16,162
1988 35,447 20,868
1989 42,674 27,591
1990 48,634 31,335
1991 59,660 36,560
1992 78,530 41,055
1993 78,834 44,730
1994 71,874 49,095
1995 68,505 49,456
1996 59,347 38,510
1997 47,149 20,736
1998 38,393 19,005
1999 25,174 18,454
2000 25,522 17,347
2001 25,643 17,402
2002 26,464 16,371
Total 802,118 489,093
Table 12.12 Adults and Adolescents only, United States
9.

Use the columns "year" and "# flu cases diagnosed. Why is “year” the independent variable and “# flu cases diagnosed.” the dependent variable (instead of the reverse)?


Use the following information to answer the next two exercises. A specialty cleaning company charges an equipment fee and an hourly labor fee. A linear equation that expresses the total amount of the fee the company charges for each session is y = 50 + 100x.

10.

What are the independent and dependent variables?

11.

What is the y-intercept and what is the slope? Interpret them using complete sentences.


Use the following information to answer the next three questions. Due to erosion, a river shoreline is losing several thousand pounds of soil each year. A linear equation that expresses the total amount of soil lost per year is y = 12,000x.

12.

What are the independent and dependent variables?

13.

How many pounds of soil does the shoreline lose in a year?

14.

What is the y-intercept? Interpret its meaning.


Use the following information to answer the next two exercises. The price of a single issue of stock can fluctuate throughout the day. A linear equation that represents the price of stock for Shipment Express is y = 15 – 1.5x where x is the number of hours passed in an eight-hour day of trading.

15.

What are the slope and y-intercept? Interpret their meaning.

16.

If you owned this stock, would you want a positive or negative slope? Why?

12.2 Scatter Plots

17.

Does the scatter plot appear linear? Strong or weak? Positive or negative?

This is a scatterplot with several points plotted in the first quadrant. The points form a clear pattern, moving upward to the right. The points do not line up , but the overall pattern can be modeled with a line.
Figure 12.26
18.

Does the scatter plot appear linear? Strong or weak? Positive or negative?

This is a scatterplot with several points plotted in the first quadrant. The points move downward to the right. The overall pattern can be modeled with a line, but the points are widely scattered.
Figure 12.27
19.

Does the scatter plot appear linear? Strong or weak? Positive or negative?

This is a scatter plot with several points plotted all over the first quadrant. There is no pattern.
Figure 12.28

12.3 The Regression Equation

Use the following information to answer the next five exercises. A random sample of ten professional athletes produced the following data where x is the number of endorsements the player has and y is the amount of money made (in millions of dollars).

x y x y
0 2 5 12
3 8 4 9
2 7 3 9
1 3 0 3
5 13 4 10
Table 12.13
20.

Draw a scatter plot of the data.

21.

Use regression to find the equation for the line of best fit.

22.

Draw the line of best fit on the scatter plot.

23.

What is the slope of the line of best fit? What does it represent?

24.

What is the y-intercept of the line of best fit? What does it represent?

25.

What does an r value of zero mean?

26.

When n = 2 and r = 1, are the data significant? Explain.

27.

When n = 100 and r = -0.89, is there a significant correlation? Explain.

12.4 Testing the Significance of the Correlation Coefficient

28.

When testing the significance of the correlation coefficient, what is the null hypothesis?

29.

When testing the significance of the correlation coefficient, what is the alternative hypothesis?

30.

If the level of significance is 0.05 and the p-value is 0.04, what conclusion can you draw?

12.5 Prediction

Use the following information to answer the next two exercises. An electronics retailer used regression to find a simple model to predict sales growth in the first quarter of the new year (January through March). The model is good for 90 days, where x is the day. The model can be written as follows:

ŷ = 101.32 + 2.48x where ŷ is in thousands of dollars.

31.

What would you predict the sales to be on day 60?

32.

What would you predict the sales to be on day 90?


Use the following information to answer the next three exercises. A landscaping company is hired to mow the grass for several large properties. The total area of the properties combined is 1,345 acres. The rate at which one person can mow is as follows:

ŷ = 1350 – 1.2x where x is the number of hours and ŷ represents the number of acres left to mow.

33.

How many acres will be left to mow after 20 hours of work?

34.

How many acres will be left to mow after 100 hours of work?

35.

How many hours will it take to mow all of the lawns? (When is ŷ = 0?)

Table 12.14 contains real data for the first two decades of flu cases reporting.

Year # flu cases diagnosed # flu deaths
Pre-1981 91 29
1981 319 121
1982 1,170 453
1983 3,076 1,482
1984 6,240 3,466
1985 11,776 6,878
1986 19,032 11,987
1987 28,564 16,162
1988 35,447 20,868
1989 42,674 27,591
1990 48,634 31,335
1991 59,660 36,560
1992 78,530 41,055
1993 78,834 44,730
1994 71,874 49,095
1995 68,505 49,456
1996 59,347 38,510
1997 47,149 20,736
1998 38,393 19,005
1999 25,174 18,454
2000 25,522 17,347
2001 25,643 17,402
2002 26,464 16,371
Total 802,118 489,093
Table 12.14 Adults and Adolescents only, United States
36.

Graph “year” versus “# flu cases diagnosed” (plot the scatter plot). Do not include pre-1981 data.

37.

Perform linear regression. What is the linear equation? Round to the nearest whole number.

38.

Find the correlation coefficient.

  1. r = ________
39.

Solve.

  1. When x = 1985, ŷ = _____
  2. When x = 1990, ŷ =_____
  3. When x = 1970, ŷ =______ Why doesn’t this answer make sense?
40.

Does the line seem to fit the data? Why or why not?

41.

What does the correlation imply about the relationship between time (years) and the number of diagnosed flu cases reported in the U.S.?

42.

Plot the two given points on the following graph. Then, connect the two points to form the regression line.

Blank graph with horizontal and vertical axes.
Figure 12.29

Obtain the graph on your calculator or computer.

43.

Write the equation: ŷ= ____________

44.

Hand draw a smooth curve on the graph that shows the flow of the data.

45.

Does the line seem to fit the data? Why or why not?

46.

Do you think a linear fit is best? Why or why not?

47.

What does the correlation imply about the relationship between time (years) and the number of diagnosed flu cases reported in the U.S.?

48.

Graph “year” vs. “# flu cases diagnosed.” Do not include pre-1981. Label both axes with words. Scale both axes.

49.

Enter your data into your calculator or computer. The pre-1981 data should not be included. Why is that so?

Write the linear equation, rounding to four decimal places:

50.

Find the correlation coefficient.

  1. correlation = _____

12.6 Outliers

Use the following information to answer the next four exercises. The scatter plot shows the relationship between hours spent studying and exam scores. The line shown is the calculated line of best fit. The correlation coefficient is 0.69.

Figure 12.30
51.

Do there appear to be any outliers?

52.

A point is removed, and the line of best fit is recalculated. The new correlation coefficient is 0.98. Does the point appear to have been an outlier? Why?

53.

What effect did the potential outlier have on the line of best fit?

54.

Are you more or less confident in the predictive ability of the new line of best fit?

55.

The Sum of Squared Errors for a data set of 18 numbers is 49. What is the standard deviation?

56.

The Standard Deviation for the Sum of Squared Errors for a data set is 9.8. What is the cutoff for the vertical distance that a point can be from the line of best fit to be considered an outlier?

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