Descriptive statistics

Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs

Understandable Statistics, Concepts and Methods

The Boston Marathon is the oldest and best-known U.S. marathon. It covers a route from Hopkinton, Massachusetts, to downtown Boston. The distance is approximately 26 miles. The Boston Marathon web site has a wealth of information about the history of the race. In particular, the site gives the winning times for the Boston Marathon. They are all over 2 hours. The following data are the minutes over 2 hours for the winning male runners over two periods of 20 years each:
Earlier Period $\begin{array}{cccccccccc}23 & 23 & 18 & 19 & 16 & 17 & 15 & 22 & 13 & 10 \\ 18 & 15 & 16 & 13 & 9 & 20 & 14 & 10 & 9 & 12 \\ \text { Recent Period } & & & & & & & & \\ 9 & 8 & 9 & 10 & 14 & 7 & 11 & 8 & 9 & 8 \\ 11 & 8 & 9 & 7 & 9 & 9 & 10 & 7 & 9 & 9\end{array}$
(a) Make a stem-and-leaf display for the minutes over 2 hours of the winning times for the earlier period. Use two lines per stem.

Organizing Data

Stem-and-Leaf Displays

02:23

Understandable Statistics, Concepts and Methods

Driving would be more pleasant if we didn't have to put up with the bad habits of other drivers. USA Today reported the results of a Valvoline Oil Company survey of 500 drivers, in which the drivers marked their complaints about other drivers. The top complaints turned out to be tailgating, marked by $22 \%$ of the respondents; not using turn signals, marked by $19 \% ;$ being cut off, marked by $16 \% ;$ other drivers driving too slowly, marked by $11 \% ;$ and other drivers being inconsiderate, marked by $8 \% .$ Make a Pareto chart showing percentage of drivers listing each stated complaint. Could this information as reported be put in a circle graph? Why or why not?

Organizing Data

Bar Graphs, Circle Graphs, and Time-Series Graphs

01:57

Understandable Statistics, Concepts and Methods

How do college professors spend their time? The National Education Association Almanac of Higher Education gives the following average distribution of professional time allocation: teaching, $51 \% ;$ research, $16 \% ;$ professional growth, $5 \% ;$ community service, $11 \% ;$ service to the college, $11 \%$; and consulting outside the college, $6 \% .$ Make a pie chart showing the allocation of professional time for college professors.

Organizing Data

Bar Graphs, Circle Graphs, and Time-Series Graphs

Histograms, Frequency Polygons, and Time Series Graphs

66 Practice Problems

04:46

Statistics

25 randomly selected students were asked the number of movies they watched the previous week. The results are as follows:
$$\begin{array}{|l|l|l|l|}
\hline \text { Number of Movies } & \text { Frequency } & \text { Relative Frequency } & \text { Cumulative Relative Frequency } \\
\hline 0 & 5 & & \\
\hline 1 & 9 & & \\
\hline 2 & 6 & & \\
\hline 3 & 4 & & \\
\hline 4 & 1 & & \\
\hline
\end{array}$$a. Construct a histogram of the data.
b. Complete the columns of the chart.

Descriptive Statistics

03:03

Understandable Statistics, Concepts and Methods

Decimal Data The following data represent tonnes of wheat harvested each year $(1894-1925)$ from Plot 19 at the Rothamsted Agricultural Experiment Stations, England.
$\begin{array}{ccccccccc}2.71 & 1.62 & 2.60 & 1.64 & 2.20 & 2.02 & 1.67 & 1.99 & 2.34 & 1.26 & 1.31\end{array}$
$\begin{array}{cccccccccc}1.80 & 2.82 & 2.15 & 2.07 & 1.62 & 1.47 & 2.19 & 0.59 & 1.48 & 0.77 & 2.04\end{array}$
$\begin{array}{cccccccccc}1.32 & 0.89 & 1.35 & 0.95 & 0.94 & 1.39 & 1.19 & 1.18 & 0.46 & 0.70\end{array}$
(a) Multiply each data value by 100 to "clear" the decimals.
(b) Use the standard procedures of this section to make a frequency table and histogram with your whole-number data. Use six classes.
(c) Divide class limits, class boundaries, and class midpoints by 100 to get back to your original data values.

Organizing Data

Frequency Distributions, Histograms, and Related Topics

03:56

Understandable Statistics, Concepts and Methods

The Wind Mountain excavation site in New Mexico is an important archaeological location of the ancient Native American Anasazi culture. The following data represent depths (in cm) below surface grade at which significant artifacts were discovered at this site (Reference: A. I. Woosley and A. J. McIntyre, Mimbres Mogollon Archaeology, University of New Mexico Press). Note: These data are also available for download at the Companion Sites for this text.
$$\begin{array}{cccccccccc}
85 & 45 & 75 & 60 & 90 & 90 & 115 & 30 & 55 & 58 \\
78 & 120 & 80 & 65 & 65 & 140 & 65 & 50 & 30 & 125 \\
75 & 137 & 80 & 120 & 15 & 45 & 70 & 65 & 50 & 45 \\
95 & 70 & 70 & 28 & 40 & 125 & 105 & 75 & 80 & 70 \\
90 & 68 & 73 & 75 & 55 & 70 & 95 & 65 & 200 & 75 \\
15 & 90 & 46 & 33 & 100 & 65 & 60 & 55 & 85 & 50 \\
10 & 68 & 99 & 145 & 45 & 75 & 45 & 95 & 85 & 65 \\
65 & 52 & 82 & &
\end{array}$$
Use seven classes.

Organizing Data

Frequency Distributions, Histograms, and Related Topics

Measures of the Location of the Data

11 Practice Problems

02:43

Statistics for Business and Economics

J. D. Powers and Associates surveyed cell phone users in order to learn about the minutes of cell phone usage per month (Associated Press, June 2002 ). Minutes per month for a sample of 15 cell phone users are shown here.
\[
\begin{array}{rrr}
615 & 135 & 395 \\
430 & 830 & 1180 \\
690 & 250 & 420 \\
265 & 245 & 210 \\
180 & 380 & 105
\end{array}
\]
a. What is the mean number of minutes of usage per month?
b. What is the median number of minutes of usage per month?
c. What is the 85 th percentile?
d. J. D. Powers and Associates reported that the average wireless subscriber plan allows up to 750 minutes of usage per month. What do the data suggest about cell phone subscribers' utilization of their monthly plan?

Descriptive Statistics: Numerical Measures

04:31

Statistics for Business and Economics

The National Association of Colleges and Employers compiled information about annual starting salaries for college graduates by major. The mean starting salary for business administration graduates was $\$ 39,850$ (CNNMoney.com, February 15,2006 ). Samples with annual starting data for marketing majors and accounting majors follow (data are in thousands):
Marketing Majors $\begin{array}{lllllll}34.2 & 45.0 & 39.5 & 28.4 & 37.7 & 35.8 & 30.6 & 35.2 & 34.2 & 42.4\end{array}$
a. Compute the mean, median, and mode of the annual starting salary for both majors.
b. Compute the first and third quartiles for both majors.
c. Business administration students with accounting majors generally obtain the highest annual salary after graduation. What do the sample data indicate about the difference between the annual starting salaries for marketing and accounting majors?

Descriptive Statistics: Numerical Measures

10:52

Elementary Statistics a Step by Step Approach

The data shown represent the scores on a national achievement test for a group of 10th-grade students. Find the approximate percentile ranks of these scores by constructing a percentile
graph.
a. 220
b. 245
c. 276
d. 280
e. 300
$\begin{array}{ll}{\text { Score }} & {\text { Frequency }} \\ \hline 196.5-217.5 & {5} \\ {217.5-238.5} & {17} \\ {238.5-259.5} & {17} \\ {259.5-280.5} & {48} \\ {280.5-301.5} & {22} \\ {301.5-322.5} & {6}\end{array}$
For the same data, find the approximate scores that cor- respond to these percentiles.
f. 15th
g. 29th
h. 43rd
i. 65th
j. 80th

Data Description

Measures of Position

Box Plots

11 Practice Problems

04:23

Statistics Informed Decisions Using Data

Explain how to determine the shape of a distribution using the boxplot and quartiles.

Numerically Summarizing Data

The Five-Number Summary and Boxplots

04:38

Statistics Informed Decisions Using Data

The following data represent the age of U.S. presidents on their respective inauguration days (through Barack Obama).
$$\begin{equation}\begin{array}{lllllllll}42 & 47 & 50 & 52 & 54 & 55 & 57 & 61 & 64 \\\hline 43 & 48 & 51 & 52 & 54 & 56 & 57 & 61 & 65 \\\hline 46 & 49 & 51 & 54 & 55 & 56 & 57 & 61 & 68 \\\hline 46 & 49 & 51 & 54 & 55 & 56 & 58 & 62 & 69 \\\hline 47 & 50 & 51 & 54 & 55 & 57 & 60 & 64 & \\\hline\end{array}\end{equation}$$
(a) Find the five-number summary.
(b) Construct a boxplot.
(c) Comment on the shape of the distribution.

Numerically Summarizing Data

The Five-Number Summary and Boxplots

03:52

Statistics Informed Decisions Using Data

After giving a statistics exam, Professor Dang determined the following five-number summary for her class results: 60687789 98. Use this information to draw a boxplot of the exam scores.

Numerically Summarizing Data

The Five-Number Summary and Boxplots

Measures of the Center of the Data

39 Practice Problems

03:21

Statistics for Business and Economics

The following times were recorded by the quarter-mile and mile runners of a university track team (times are in minutes).
\[
\begin{array}{lrrrrr}
\text {Quarter-Mile Times:} & .92 & .98 & 1.04 & .90 & .99 \\
\text {Mile Times:} & 4.52 & 4.35 & 4.60 & 4.70 & 4.50
\end{array}
\]
After viewing this sample of running times, one of the coaches commented that the quartermilers turned in the more consistent times. Use the standard deviation and the coefficient of variation to summarize the variability in the data. Does the use of the coefficient of variation indicate that the coach's statement should be qualified?

Descriptive Statistics: Numerical Measures

01:32

Statistics for Business and Economics

A simple random sample of 40 items resulted in a sample mean of $25 .$ The population standard deviation is $\sigma=5$
a. What is the standard error of the mean, $\sigma_{\bar{x}} ?$
b. $\quad$ At $95 \%$ confidence, what is the margin of error?

Interval Estimation

02:01

Statistics Informed Decisions Using Data

For each of the following situations, determine which measure of central tendency is most appropriate and justify your reasoning.
(a) Average price of a home sold in Pittsburgh, Pennsylvania, in 2011
(b) Most popular major for students enrolled in a statistics course
(c) Average test score when the scores are distributed symmetrically
(d) Average test score when the scores are skewed right
(e) Average income of a player in the National Football League
(f) Most requested song at a radio station

Numerically Summarizing Data

Measures of Central Tendency

Skewness and the Mean, Median, and Mode

49 Practice Problems

01:45

Understandable Statistics, Concepts and Methods

Interpretation A job-performance evaluation form has these categories:
$1=$ excellent; $2=$ good; $3=$ satisfactory; $4=$ poor; $5=$ unacceptable
Based on 15 client reviews, one employee had
median rating of $4 ;$ mode rating of 1
The employee was pleased that most clients had rated her as excellent. The supervisor said improvement was needed because at least half the clients had rated the employee at the poor or unacceptable level. Comment on the different perspectives.

Averages and Variation

Measures of Central Tendency: Mode, Median, and Mean

00:52

Statistics Informed Decisions Using Data

Sketch four histograms-one skewed right, one skewed left, one bell-shaped, and one uniform. Label each histogram according to its shape. What makes a histogram skewed left? Skewed right? Symmetric?

Organizing and Summarizing Data

Organizing Quantitative Data: The Popular Displays

03:30

Elementary Statistics: Picturing the World

Song Lengths Side-by-side box-and-whisker plots can be used to compare two or more different data sets. Each box-and-whisker plot is drawn on the same number line to compare the data sets more easily. The lengths (in seconds) of songs played at two different concerts are shown.
(a) Describe the shape of each distribution. Which concert has less variation in song lengths?
(b) Which distribution is more likely to have outliers? Explain your reasoning.
(c) Which concert do you think has a standard deviation of $16.3 ?$ Explain your reasoning.
(d) Can you determine which concert lasted longer? Explain.

Descriptive Statistics

Measures of Position

Measures of the Spread of the Data

19 Practice Problems

07:26

Statistics Informed Decisions Using Data

According to the U.S. Census Bureau, the mean of the commute time to work for a resident of Boston, Massachusetts, is 27.3 minutes. Assume that the standard deviation of the commute time is 8.1 minutes to answer the following:
(a) What minimum percentage of commuters in Boston has a commute time within 2 standard deviations of the mean?
(b) What minimum percentage of commuters in Boston has a commute time within 1.5 standard deviations of the mean? What are the commute times within 1.5 standard deviations of the mean?
(c) What is the minimum percentage of commuters who have commute times between 3 minutes and 51.6 minutes?

Numerically Summarizing Data

Measures of Dispersion

01:25

Statistics Informed Decisions Using Data

You have received a yearend bonus of $\$ 5000 .$ You decide to invest the money in the stock market and have narrowed your investment options down to two mutual funds. The following data represent the historical quarterly rates of return of each mutual fund for the past 20 quarters (5 years).
Describe each data set. That is, determine the shape, center, and spread. Which mutual fund would you invest in and why?

Numerically Summarizing Data

Measures of Dispersion

01:46

Statistics Informed Decisions Using Data

Suppose that you are in the market to purchase a car. With gas prices on the rise, you have narrowed it down to two choices and will let gas mileage be the deciding factor. You decide to conduct a little experiment in which you put 10 gallons of gas in the car and drive it on a closed track until it runs out gas. You conduct this experiment 15 times on each car and record the number of miles driven.
Describe each data set. That is, determine the shape, center, and spread. Which car would you buy and why?

Numerically Summarizing Data

Measures of Dispersion

26 Practice Problems

02:36

Essentials of Statistics for Business and Economics

Figure 1.8 provides a bar chart showing the amount of federal spending for the years 2002 to $2008(\text {USA Today}, \text { February } 5,2008$ ).
a. What is the variable of interest?
b. Are the data categorical or quantitative?
c. Are the data time series or cross-sectional?
d. Comment on the trend in federal spending over time.

Data and Statistics

02:43

Statistics for Business and Economics

Consider a sample with data values of $27,25,20,15,30,34,28,$ and $25 .$ Compute the $20 \mathrm{th}$ $25 \mathrm{th}, 65 \mathrm{th},$ and 75 th percentiles.

Descriptive Statistics: Numerical Measures

0:00

Mathematical Statistics with Applications

Refer to Exercise $9.66 .$ Suppose that a sample of size $n$ is taken from a normal population with mean $\mu$ and variance $\sigma^{2} .$ Show that $\sum_{i=1}^{n} Y_{i},$ and $\sum_{i=1}^{n} Y_{i}^{2}$ jointly form minimal sufficient statistics for $\mu$ and $\sigma^{2}$.

Properties of Point Estimators and Methods of Estimation

The Rao-Blackwell Theorem and Minimum-Variance Unbiased nation

Coefficient of Variation

11 Practice Problems

01:32

Elementary Statistics: Picturing the World

Find the coefficient of variation for each of the two data sets. Then compare the results.
The ages (in years) and weights (in pounds) of all wide receivers for the 2012 San Diego Chargers are listed.
$$\begin{aligned}
&\begin{array}{lllllllll}
\text { Yes } & 25 & 24 & 24 & 31 & 25 & 28 & 26 & 30 & 22
\end{array}\\
&\text { Weights } 215 \quad 217 \quad 190 \quad 225 \quad 192 \quad 215 \quad 185 \quad 210 \quad 220
\end{aligned}$$

Descriptive Statistics

Measures of Variation

01:48

Elementary Statistics: Picturing the World

Find the coefficient of variation for each of the two data sets. Then compare the results.
Sample batting averages for baseball players from two opposing teams are listed.
$$\begin{array}{llllll}
\text { Team A } & 0.295 & 0.310 & 0.325 & 0.272 & 0.256 \\
& 0.297 & 0.320 & 0.384 & 0.235 & 0.297
\end{array}$$
$$\begin{array}{llllll}
\text { Team B } & 0.223 & 0.312 & 0.256 & 0.300 & 0.238 \\
& 0.299 & 0.204 & 0.226 & 0.292 & 0.260
\end{array}$$

Descriptive Statistics

Measures of Variation

01:44

Elementary Statistics: Picturing the World

Find the coefficient of variation for each of the two data sets. Then compare the results.
The ages (in years) and heights (in inches) of all pitchers for the 2013 St. Louis Cardinals are listed.
$$\begin{array}{lllllllllllll}
\text { Ages } & 24 & 29 & 37 & 24 & 26 & 25 & 24 & 32 & 22 & 29 & 23 & 31 \\
\text { Heights } & 72 & 76 & 73 & 73 & 77 & 76 & 72 & 74 & 75 & 75 & 74 & 79
\end{array}$$

Descriptive Statistics

Measures of Variation

Quartiles, Interquartile Range, Outliers and z-score

44 Practice Problems

01:09

Statistics Informed Decisions Using Data

Explain what each quartile represents.

Numerically Summarizing Data

Measures of Position and Outliers

01:04

Statistics Informed Decisions Using Data

Explain the circumstances for which the interquartile range is the preferred measure of dispersion. What is an advantage that the standard deviation has over the interquartile range?

Numerically Summarizing Data

Measures of Position and Outliers

02:35

Statistics Informed Decisions Using Data

Explain the advantage of using $z$ -scores to compare observations from two different data sets.

Numerically Summarizing Data

Measures of Position and Outliers

Box-and-Whisker Plot

17 Practice Problems

02:50

Understandable Statistics, Concepts and Methods

Some data sets include values so high or so low that they seem to stand apart from the rest of the data. These data are called outliers. Outliers may represent data collection errors, data entry errors, or simply valid but unusual data values. It is important to identify outliers in the data set and examine the outliers carefully to determine if they are in error. One way to detect outliers is to use a box-and-whisker plot. Data values that fall beyond the limits,
$$\begin{aligned}
&\text { Lower limit: } Q_{1}-1.5 \times(I Q R)\\
&\text { Upper limit: } Q_{3}+1.5 \times(I Q R)
\end{aligned}$$
where $I Q R$ is the interquartile range, are suspected outliers. In the computer software package Minitab, values beyond these limits are plotted with asterisks (*). Students from a statistics class were asked to record their heights in inches. The heights (as recorded) were $$\begin{array}{cccccccccccc}
65 & 72 & 68 & 64 & 60 & 55 & 73 & 71 & 52 & 63 & 61 & 74 \\
69 & 67 & 74 & 50 & 4 & 75 & 67 & 62 & 66 & 80 & 64 & 65
\end{array}$$ (a) Make a box-and-whisker plot of the data.
(b) Find the value of the interquartile range $(I Q R)$
(c) Multiply the IQR by 1.5 and find the lower and upper limits.
(d) Are there any data values below the lower limit? above the upper limit? List any suspected outliers. What might be some explanations for the outliers?

Averages and Variation

Percentiles and Box-and-Whisker Plots

01:14

Understandable Statistics, Concepts and Methods

Angela took a general aptitude test and scored in the $82 n d$ percentile for aptitude in accounting. What percentage of the scores were at or below her score? What percentage were above?

Averages and Variation

Percentiles and Box-and-Whisker Plots

06:57

Elementary Statistics: Picturing the World

Modified Boxplot $A$ modified boxplot is a boxplot that uses symbols to identify outliers. The horizontal line of a modified boxplot extends as far as the minimum data entry that is not an outlier and the maximum data entry that is not an outlier. In Exercises 57 and $58,(a)$ identify any outliers and (b) draw a modified boxplot that represents the data set. Use asterisks (*) to identify outtliers.
$$\begin{array}{rrrrrrrrrrrr}
75 & 78 & 80 & 75 & 62 & 72 & 74 & 75 & 80 & 95 & 76 & 72
\end{array}$$

Descriptive Statistics

Measures of Position

Percentage and Percentile

25 Practice Problems

01:23

Statistics Informed Decisions Using Data

Suppose you received the highest score on an exam. Your friend scored the second-highest score, yet you both were in the 99th percentile. How can this be?

Numerically Summarizing Data

Measures of Position and Outliers

01:04

Statistics Informed Decisions Using Data

Write a paragraph that explains the meaning of percentiles.

Numerically Summarizing Data

Measures of Position and Outliers

02:53

Statistics Informed Decisions Using Data

The following graph is an ogive of the mathcmatics scorcs on the SAT for the class of 2010. The vertical axis in an ogive is the cumulative relative frequency and can also be interpreted as a percentile.
(a) Find and interpret the percentile rank of a student who scored 450 on the SAT mathematics exam.
(b) Find and interpret the percentile rank of a student who scored 750 on the SAT mathematics exam.
(c) If Jane scored at the 44 th percentile, what was her score?

Numerically Summarizing Data

Measures of Position and Outliers

Histogram

4 Practice Problems

00:10

Elementary Statistics

When using histograms to compare two data sets, it is sometimes difficult to make comparisons by looking back and forth between the two histograms. A back-to-back relative frequency histog ram has a format that makes the comparison much easier. Instead of frequencies, we should use relative frequencies (percentages or proportions) so that the comparisons are not difficult when there are different sample sizes. Use the relative frequency distributions of the ages of Oscarwinning actresses and actors from Exercise 19 in Section 2 - 1 on page $49,$ and complete the back-to-back relative frequency histograms shown below. Then use the result to compare the two data sets.
(GRAPH CAN'T COPY)

Exploring Data with Tables and Graphs

Histograms

00:14

Elementary Statistics

Answer the questions by referring to the following Minitab-generated histogram, which depicts the weights (grams) of all quarters listed in Data Set 29 "Coin Weights" in Appendix $B$. (Grams are actually units of mass and the values shown on the horizontal scale are rounded.)
How would the shape of the histogram change if the vertical scale uses relative frequencies expressed in percentages instead of the actual frequency counts as shown here?
(GRAPH CAN'T COPY)

Exploring Data with Tables and Graphs

Histograms

00:35

Elementary Statistics

If we collect a sample of blood platelet counts much larger than the sample included with Exercise $3,$ and if our sample includes a single outlier, how will that outlier appear in a histogram?

Exploring Data with Tables and Graphs

Histograms

Root Mean Square

3 Practice Problems

0:00

Mathematical Statistics with Applications

Move down to the portion of the applet labeled "Curvilinear Relationship" associated with the applet Fitting a line Using Least Squares.
a. Does it seem like a straight line will provide a good fit to the data in the graph? Does it seem that there is likely to be some functional relationship between $E(Y)$ and $x ?$
b. Is there any straight line that fits the data better than the one with 0 slope?
c. If you fit a line to a data set and obtain that the best fitting line has 0 slope, does that mean that there is no functional relationship between $E(Y)$ and the independent variable? Why?

Linear Models and Estimation by Least Squares

The Method of Least Squares

01:33

Mathematical Statistics with Applications

If $\widehat{\beta}_{0}$ and $\widehat{\beta}_{1}$ are the least-squares estimates for the intercept and slope in a simple linear regression model, show that the least-squares equation $\hat{y}=\hat{\beta}_{0}+\hat{\beta}_{1} x$ always goes through the point $(\bar{x}, \bar{y})$. [Hint: Substitute $\bar{x}$ for $x$ in the least-squares equation and use the fact that $\left.\widehat{\beta}_{0}=\bar{y}-\widehat{\beta}_{1} \bar{x} .\right]$

Linear Models and Estimation by Least Squares

The Method of Least Squares

00:48

Statistics

The r.m.s. error of the regression line for predicting $y$ from $x$ is
$\begin{array}{ll}{\text { (i) }} & {\text { SD of } y} & {\text { (iv) } r \times \text { SD of } x} \\ {\text { (ii) }} & {\text { SD of } x} & {\text { (v) } \sqrt{1-r^{2}} \times \text { SD of } y} \\ {\text { (iii) } r \times \operatorname{SD} \text { of } y} & {\text { (vi) } \sqrt{1-r^{2}} \times \text { SD of } x}\end{array}$

The R.M.S. Error for Regression

Errors and Outliers

5 Practice Problems

01:49

STATS Modeling The World

Elephants and hippos We removed humans from the scatterplot in Exercise 31 because our species was an outlier in life expectancy. The resulting scatterplot shows two points that now may be of concern. The point in the upper right corner of this scatterplot is for elephants, and the other point at the far right is for hippos.
a) By removing one of these points, we could make the association appear to be stronger. Which point? Explain.
b) Would the slope of the line increase or decrease?
c) Should we just keep removing animals to increase the strength of the model? Explain.
d) If we remove elephants from the scatterplot, the slope of the regression line becomes 11.6 days per year. Do you think elephants were an influential point? Explain.

Exploring Relationships Between Variables

Regression Wisdom

01:47

STATS Modeling The World

Swim the lake 2010 People swam across Lake Ontario 48 times between 1974 and 2010 (www.soloswims.com). We might be interested in whether they are getting any faster or slower. Here are the regression of the crossing Times (minutes) against the Year of the crossing and the residuals plot:
a) What does the $R^{2}$ mean for this regression?
b) Are the swimmers getting faster or slower? Explain.
c) The outlier seen in the residuals plot is a crossing by Vicki Keith in 1987 in which she swam a round trip, north to south, and then back again. Clearly, this swim doesn't belong with the others. Would removing it change the model a lot? Explain.

Exploring Relationships Between Variables

Regression Wisdom

03:35

STATS Modeling The World

Tracking hurricanes 2010 In a previous chapter, we saw data on the errors (in nautical miles) made by the National Hurricane Center in predicting the path of hurricanes. The scatterplot below shows the trend in the 24-hour tracking errors since 1970 (www.nhc.noaa.gov).
a) Interpret the slope and intercept of the model.
b) Interpret se in this context.
c) The Center would like to achieve an average tracking error of 45 nautical miles by 2015. Will they make it? Defend your response.
d) What if their goal were an average tracking error of 25 nautical miles?
e) What cautions would you state about your conclusion?

Exploring Relationships Between Variables

Regression Wisdom

Chebyshev’s Inequality

2 Practice Problems

04:17

Statistics Informed Decisions Using Data

A popular theory in investment states that you should invest a certain amount of money in foreign investments to reduce your risk. The risk of a portfolio is defined as the standard deviation of the rate of return. Refer to the graph on the following page, which depicts the relation between risk (standard deviation of rate of return) and reward (mean rate of return).
(a) Determine the average annual return and level of risk in a portfolio that is $10 \%$ foreign.
(b) Determine the percentage that should be invested in foreign stocks to best minimize risk.
(c) Why do you think risk initially decreases as the percent of foreign investments increases?
(d) A portfolio that is $30 \%$ foreign and $70 \%$ American has a mean rate of return of about $15.8 \%,$ with a standard deviation of $14.3 \% .$ According to Chebyshev's Inequality, at least $75 \%$ of returns will be between what values? According to Chebyshev's Inequality, at least $88.9 \%$ of returns will be between what two values? Should an investor be surprised if she has a negative rate of return? Why?

Numerically Summarizing Data

Measures of Dispersion

0:00

Statistics Informed Decisions Using Data

Chebyshev's Inequality applies to all distributions regardless of shape, but the Empirical Rule holds only for distributions that are bell shaped.

Numerically Summarizing Data

Measures of Dispersion

Variance, Standard Deviation, and Range

11 Practice Problems

01:35

Statistics Informed Decisions Using Data

Which of the following would have a higher standard deviation? (a) IQ of students on your campus or (b) IQ of residents in your home town? Why?

Numerically Summarizing Data

Measures of Dispersion

02:38

Statistics Informed Decisions Using Data

Are any of the measures of dispersion mentioned in this section resistant? Explain.

Numerically Summarizing Data

Measures of Dispersion

01:53

Statistics Informed Decisions Using Data

Suppose Professor Alpha and Professor Omega each teach Introductory Biology. You need to decide which professor to take the class from and have just completed your Introductory Statistics course. Records obtained from past students indicate that students in Professor Alpha's class have a mean score of $80 \%$ with a standard deviation of $5 \%,$ while past students in Professor Omega's class have a mean score of $80 \%$ with a standard deviation of $10 \% .$ Decide which instructor to take for Introductory Biology using a statistical argument.

Numerically Summarizing Data

Measures of Dispersion

19 Practice Problems

03:36

Statistics Informed Decisions Using Data

Use the following steps to approximate the median from grouped data.
Step 1 Construct a cumulative frequency distribution.
Step 2 Identify the class in which the median lies. Remember, the median can be obtained by determining the observation that lies in the middle.
Step 3 Interpolate the median using the formula
$$\text { Median }=M=L+\frac{\frac{n}{2}-C F}{f}(i)$$
where $L$ is the lower class limit of the class containing the median $n$ is the number of data values in the frequency distribution CF is the cumulative frequency of the class immediately preceding the class containing the median $f$ is the frequency of the median class $i$ is the class width of the class containing the median.
Approximate the median of the frequency distribution in Problem 4.

Numerically Summarizing Data

Measures of Central Tendency and Dispersion rouped Data

04:56

Statistics Informed Decisions Using Data

The data on the next page represent the age of the mother at childbirth for 1980 and 2007 . (TABLE CAN'T COPY)
(a) Approximate the population mean and standard deviation of age for mothers in 1980 .
(b) Approximate the population mean and standard deviation of age for mothers in 2007
(c) Which year has the higher mean age?
(d) Which year has more dispersion in age?

Numerically Summarizing Data

Measures of Central Tendency and Dispersion rouped Data

06:57

Statistics Informed Decisions Using Data

The following data represent the male and female population, by age, of the United States in 2008 Note: Use 95 for the class midpoint of $\geq 90 .$
$$\begin{array}{cc}\text { Age } & \begin{array}{c}\text { Male Resident Pop } \\\text { (in thousands) }\end{array} & \begin{array}{c}\text { Female Resident } \\\text { Pop (in thousands) }\end{array} \\\hline 0-9 & 929 & 19,992 \\
\hline 10-19 & 21,074 & 20,278 \\\hline 20-29 & 21,105 & 20,482 \\\hline 30-39 & 19,780 & 20,042 \\\hline 40-49 & 21,754 & 22,346 \\\hline 50-59 & 19,303 & 20,302 \\\hline 60-69 & 12,388 & 13,709 \\\hline 70-79 & 6940 & 8837 \\\hline 80-89 & 3106 & 9154 \\\hline \geq 90 & 479 & 1263 \\\hline \text { Source: U.S. Census Bureau } &
\end{array}$$
(a) Approximate the population mean and standard deviation of age for males.
(b) Approximate the population mean and standard deviation of age for females.
(c) Which gender has the higher mean age?
(d) Which gender has more dispersion in age?

Numerically Summarizing Data

Measures of Central Tendency and Dispersion rouped Data

Five-Number Summary

3 Practice Problems

01:02

Statistics Informed Decisions Using Data

What does the five-number summary consist of?

Numerically Summarizing Data

The Five-Number Summary and Boxplots