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Unformatted text preview: 68 Chapter 3 Some Basic Concepts of Statistics formal schooling. Most workers are on the job for approximately 40 years before they
retire. There are approximately four people in a typical family, and that family most
likely has two cars. On a summer day, the temperature will be around 80 degrees Fahrenm
heit, on a winter day around 35 degrees Fahrenheit. The winnin g footbali team may score
around 21 points, the winning basketball team around 90 points, and the winning
baseball team around 5 runs. The newborn baby weighs approximately 8 pounds. So, you
see, all of us have a set of “typical” values by which we make judgments every day. Dis'
cuss a variety of typical vaiues that are useful in your life. Where did you learn these valm
ues? Do you think the values are correct, or nearly so? lgiﬁﬁow can we choose one brand of sports drinic over another? Perhaps knowing the number
MW ' of calories and price will help. The data that follow show this information for the leading liquid Sports drinks. One Swounce serving is the basic unit for both calories and cost. Brand ‘ Caiories (C) Cost in dollars (D)
lGK 60 0.22
All Sport 70 0.24
Daily’s Est Ade 60 0.26
Exceed 70 0.34
Gatorade 50 0.26
Hydra Fuel 66 0.52
Nautilus Plus . 60 0.22 . 3 ‘ Power Ade 67 0.24 1 : Snapple SnapUp 80 0.35 SOURCE: Consumer Reports, August i993. _ J ‘ a. What is a good summary number for typical caiories per serving for these drinks?
. 3 ' What is a good summary number for the variation in the calories per serving?
i t I b. What is a good summary number for typical cost per serving for these drinks? What is a good summary number for the variation in costs per serving? c. Does the total of the calories column provide useful information? How about the total
of the cost column? 4 6. Suppose Hydra Fuel is eliminated from the list. What impact does that have on the
average calories per serving? On the standard deviation of calories per serving? On
the average cost per serving? 0n the standard deviation of the cost per serving? e. Which drink has the most inﬂuence on the average calories per serving? What rea—
soning did you use in matting this choice? 3.11 Some of the sports drinks come in powdered form or in “light" versions.
a. Exceed powder comes in a 32serving container for $9.43 and has '70 calories/
serving. Gatorade powder has a 32serving size for $3.59 and has 60 calories/serving. Is it fair to include the powdered drinks on the same list with the liquid drinks and 1 ﬁgure their values into the averages? Why or why not? b. All Sport Lite has 2 calories/serving, at a cost of 24¢ a serving. Gatorade Light has
25 calories/serving, at a cost of 26¢ a serving. Will adding the light varieties to the list
have much of an effect on the average cost per serving? Will adding the light varieties have much of an effect on the standard deviation of cost per serving? ; before they
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ght varieties Exercises 69 c. Will adding the light varieties have much of an effect on the average calories per serv
ing? Will adding the light varieties have much of an effect on the standard deviation
of calories per serving? Describe the nature of this effect. d. Can you think of a way to choose a typical value for the calories that is less affected m» by the two low values for the light varieties?
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If Slinany animals are in danger of extinction. One way to see the extent of the problem is to
N...» study the numbers of animals on the endangered species list. The following data show
the numbers of endangered species for various groups of animals. The count is the total
number of endangered species within the group. Thus, there are 37 mammals on the
endangered species list within the United States and 249 others in the rest of the world. Group U.S. US. and foreign Foreign only
Mammals 37 19 . 249
Birds 57 16 15 3
Reptiles 8 8 64
Amphibians 6 O 8
Fishes 55 3 l I
Snails 12 0 I
Clams 50 0 2
Crustaceans 10 {l 0
Insects 13 2 4
Arachnids 3 0 0 SOURCE: The WorldAImanac. 1994. a. If you wanted to summarize these data for the United States in a single number, what
number do you think would be the most meaningfui? Why? in. If you wanted to summarize the situation for endangered mammals worldwide,
including the United States, what number do you think would be the most meaning
ful? Why? ' c. Does the average of the numbers in the United States column have a useful interpre—
tation? Explain. d. Write a paragraph summarizing the information in the data set. Matte use of the sum
mary numbers you chose in parts (a) and (b). 3.13 When a few data points are repeated in a data set, the results are often arrayed in a fre— quency table. For example, a quiz given to each of 25 students was graded on a four»
point scale (0, 1, 2, 3) with 3 being a perfect score. Here are the results. WWW
Score (X) Frequency (F) Proportion (P) 3 16 0.64
2 4 0.16
l 2 0.08
0 3 0.12 M“ m._..w_h 1“ a,“ 70 Chapter 3 Some Basic Concepts of Statistics 3. Show how the average score can be calculated by using the frequencies.
b. Show how the average score can be calculated try using the proportions. c. Calculate the standard deviation of these scores. 3.14 According to the U. 8. Census Bureau, the distribution of family sizes in the United 'States for the year 200615 as shown on the accompanying table. (The tennfamz‘ly refers to a group of two or more people related by birth, marriage, or adoption and residing to
gether in a household. The term household refers to all people who occupy a housing
unit, that is, a house, an apartment or other group of rooms, or a single room that consti—
tutes separate living quarters.) Size of family Number of families Percentage of
(2000) (thousands) families 2 31,455 . 44.3 3 16,973 22.9 4 14,496 20.0 5 65 26 8.6 6 2226 2.8 7* 3249 1.4 *This category is actually “7 or more," but very few families have more than
seven members. That is, 44.3% of the families in the United States have two members, whereas only
2.8% have six members. {Families with more than seven members are very rare.) a. h.
c. Find the mean family size, approximately, from this distribation of family sizes. Will
this approximation be too large or too small? Explain. Find the approximate standard deviation of the family sizes. Suppose Nielsen randomly selects 400 families from this population. Describe, as closely as you can, the shape, center, and spread of the 400 data values that might
occur in the sample. . Nielsen is actually interested in the mean number of persons per family in samples of 400 families. Describe, as closely as you can, the shape, center, and Spread of
the distribution of possible values of the sample mean in random samples of 400
families. 3.15 The table here gives the average SAT score for the 20024003 school year for each state,
along with the percentage of high school seniors who took the test. 3. Plot the points with the percentage on the horizontal axis and the average score on the
vertical axis. Describe the relationship between these two variables and suggest rea~
sons for the pattern you see. . Guess a vaiue for the correlation coefﬁcient here. If possible, check your guess against the calculated correlation coefﬁcient. Does correlation appear to be a good
measure of the strength of the relationship between these two variables? Explain your
answer. ...
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