Chapter 3 Examples
Example 1: Math Scores (page 85)
The figure below shows the distribution of math achievement scores for 46 countries as determined by the National Assessment of Education Progress. The scores are meant to measure the accomplishment of each country’s eighth grade students with respect to mathematical achievement.
Example 2: Gas Buddy (page 86)
According to GasBuddy.com (a website that invites people to submit prices at local gas stations), the prices of 1 gallon of regular gas at 12 service stations in a neighborhood in Austin, Texas, were as follows on one fall day in 2013:
A dot plot (not shown) indicates that the distribution is fairly symmetrical.
Find the mean price of a gallon of regular gas at these service stations. Explain what the value of the mean signifies in this context (in other words, interpret the mean).
Example 3: Mean Smog Levels (page 87)
The histogram below shows the distribution of the amount of particulate level, or smog, in the air in 333 cities in the United States in 2008, as reported by the Environmental Protection Agency (EPA). The units are micrograms of particles per cubic meter. When inhaled, these particles can affect the heart and lungs, so you would prefer your city to have a fairly low amount of particulate matter. Looking at the histogram, estimate the mean value of particles for those 333 cities.
We used four different statistical software programs to find the mean particulate level in the 333 cities. The data were uploaded into StatCrunch, Minitab, Excel, and the TI-84 calculator. For each of the computer outputs shown below, find the mean particulate matter.
StatCrunch:
Minitab:
TI-84:
Excel:
Standard Deviation Example
The two histograms below show daily high temperatures in degrees Fahrenheit recorded over one recent year in two locations in the United States, Provo, Utah and San Francisco, California.
Compare their means and standard deviations.
Example 4: Comparing Standard Deviations from Histograms (page 90)
Each of the graphs below shows a histogram for a distribution of the same number of observations, and all of the distributions have a mean of about 3.5
Which distribution has the largest standard deviation, and why?
Example 5: Standard Deviation of Smog Levels (page 91)
The mean particulate matter in the 333 cities in example 3 is 10.7 micrograms per cubic meter, and the standard deviation is 2.6 micrograms per cubic meter.
Find the level of particulate matter one standard deviation above the mean and one standard deviation below the mean. Keeping in mind that the EPA says that levels over 15 micrograms per cubic meter are unsafe, what can we conclude about the air quality of most of the cities in this sample?
Example 6: A Gallon of Gas (page 92)
From the website GasBudy.com, we collected the prices of a gallon of regular gas at 12 gas stations in a neighborhood in Austin, Texas, for one day in October 2013.
Find the standard deviation for the prices. Explain what this value means in the context of the data.
Example 7: Comparing the Empirical Rule to Actual Smog Levels (page 93)
The mean particulate matter in the 333 cities in example 3 is 10.7 micrograms per cubic meter, and the standard deviation is 2.6 micrograms per cubic meter. Because this distribution is roughly unimodal and symmetric the Empirical Rule can be applied.
About 68% of the cities will have PM levels between what two values?
About 95% of the cities will have PM levels between what two values?
Nearly all of the cities will have PM levels between what two values?
The histograms below show the actual distribution of values. The location of the mean is indicated, as well as the boundaries for points within one (a), two (b), and three (c) standard deviations. Compare the actual values to those predicted by the Empirical Rule.
Example 8: Temperatures in San Francisco (page 96)
The mean daily high temperature in San Francisco is 65˚F and the standard deviation is 8˚. Using the Empirical Rule, decide whether it is unusual in San Francisco to have a day when the maximum high temperature is colder than 49˚F.
Example 9: Exam Scores (page 97)
Maria scored 80 out of 100 on her first stats exam in a course and 85 out of 100 on her second stats exam. On the first exam, the mean was 70 and the standard deviation was 10. On the second exam, the mean was 80 and the standard deviation was 5. On which exam did Maria perform better when compared to the whole class?
Example 10: Daily Temperatures (page 98)
The mean daily high temperature in San Francisco is 65˚F and the standard deviation is 8˚F. On one day, the high temperature was 49˚F. What is this temperature in standard units (the z-score)? Assuming the Empirical Rule applies, does this seem unusual?
Example 11: Twelve Gas Stations (page 101)
The prices of a gallon of regular gas at 12 Austin, Texas gas stations in October 2013 (see ex. 6) were:
Determine the median price for a gallon of gas and interpret the value.
Example 12: Sliced Turkey (page 102)
The dotplot below shows that the median percentage of fat from various brands of sliced ham for sale at a grocery store was 23.5%.
Here are percentages of fat in sliced turkey available at the same store.
How do the median percentages compare?
Example 13: Heights of Children (page 104)
A group of eight children have the following heights (in inches):
Find the interquartile range for the children’s height
Example 14: MP3 Song Lengths (page 107)
One of the authors of your textbook created a data set of the songs on his mp3 player. A histogram of the data is shown below.
He wants to describe the distribution of song lengths. What measures should he use for the center and spread: the mean (250.2 seconds) with the standard deviation (152.0 seconds) or the median (226 seconds) with the interquartile range (117 seconds)? Interpret the appropriate measures.
Example 15: Fast Food (page 108)
A (very small) fast-food restaurant has five employees, all of whom work full-time for $7 per hour. Each employee’s annual income is about $16,000 per year. The owner, on the other hand, makes $100,000 per year.
Calculate both the mean and the median. Which would you use to represent the typical income at this business – the mean or the median? Why?
Example 16: Marathon Times, Revisited (page 110)
The following graphs show the same data, separated into groups, Olympic athletes and amateurs.
Why was median used to indicate typical finishing time?
Example 17: Skyscraping (page 112)
Sketch the boxplot. Describe how you determined where to draw the whiskers. Are there any outliers? Are the outliers mostly short buildings or very tall buildings?