Download Elementary Statistics Using Excel Mario F. Triola Fifth Edition

Elementary Statistics Using Excel
Triola
5e
ISBN 978-1-29204-176-6
9 781292 041766
Elementary Statistics Using Excel
Mario F. Triola
Fifth Edition
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ISBN 10: 1-292-04176-5
ISBN 10: 1-269-37450-8
ISBN 13: 978-1-292-04176-6
ISBN 13: 978-1-269-37450-7
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Printed in the United States of America
Discrete Probability Distributions
answer on the probability of exactly 501 heads, which is the small value of 0.0252.
In this situation, any specific number of heads will have a very low probability.)
Based on the 0.487 probability of getting 501 heads or more, we conclude that
501 heads in 1000 tosses of a fair coin is not an unusually high number of heads.
(See Example 11 in Section 2.)
Example 7
Identifying Unusual Results with Probabilities
The Chapter Problem includes results consisting of 879 girls in 945 births. Is 879
girls in 945 births an unusually high number of girls? What does it suggest about
the effectiveness of the XSORT method of gender selection?
Solution
The result of 879 girls in 945 births is more than we expect under normal circumstances, so we want to determine whether 879 girls is unusually high. Here, the
relevant probability is the probability of getting 879 or more girls in 945 births.
Using methods covered later in Section 3, we can find that P (879 or more girls in
945 births) = 0.000 when rounded to three decimal places. (We can denote such
a probability as 0 + .) Because the probability of getting 879 or more girls is less
than or equal to 0.05, we conclude that 879 girls in 945 births is an unusually high
number of girls.
Interpretation
Because it is so unlikely that we would get 879 or more girls in 945 births by
chance, these results suggest that the XSORT method of gender selection is
effective in increasing the likelihood that a baby will be a girl. (However, this
does not prove that the XSORT method is responsible for the large number
of girls.)
Part 2: Expected Value in Decision Theory and Rationale for Formulas 1
through 4
Expected Value in Decision Theory
In Part 1 of this section we noted that the expected value of a random variable x
is equal to the mean m. We can therefore find the expected value by computing
Σ 3 x # P (x) 4 , just as we do for finding the value of m. We also noted that the concept
of expected value is used in decision theory. In Example 8 we illustrate this use of
expected value with a situation in which we must choose between two different bets.
Example 8 involves a real and practical decision.
Example 8
Be a Better Bettor
You have $5 to place on a bet in the Golden Nugget casino in Las Vegas. You have
narrowed your choice to one of two bets:
Roulette: Bet on the number 7 in roulette.
Craps: Bet on the “pass line” in the dice game of craps.
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Discrete Probability Distributions
a.
If you bet $5 on the number 7 in roulette, the probability of losing $5 is
37>38 and the probability of making a net gain of $175 is 1>38. (The prize
is $180, including your $5 bet, so the net gain is $175.) Find your expected
value if you bet $5 on the number 7 in roulette.
b.
If you bet $5 on the pass line in the dice game of craps, the probability of
losing $5 is 251>495 and the probability of making a net gain of $5 is
244>495. (If you bet $5 on the pass line and win, you are given $10 that
includes your bet, so the net gain is $5.) Find your expected value if you bet
$5 on the pass line.
Which of the preceding two bets is better? Why?
Solution
a.
Roulette The probabilities and payoffs for betting $5 on the number 7 in
roulette are summarized in Table 5. Table 5 also shows that the expected
value is Σ 3 x # P(x)4 = -26c. That is, for every $5 bet on the number 7,
you can expect to lose an average of 26c.
Table 5 Roulette
Event
Lose
Win (net gain)
x
P (x)
- $5
37/38
- $4.868421
$175
1/38
$4.605263
- $0.26 (rounded)
(or - 26¢)
Total
b.
x · P (x)
Dice The probabilities and payoffs for betting $5 on the pass line in craps
are summarized in Table 6. Table 6 also shows that the expected value is
Σ 3 x # P (x) 4 = -7c. That is, for every $5 bet on the pass line, you can
expect to lose an average of 7c.
Table 6 Dice
Event
x
P (x)
Lose
- $5
251/495
Win (net gain)
$5
244/495
Total
x · P (x)
- $2.535353
$2.464646
- $0.07 (rounded)
(or - 7¢)
Interpretation
The $5 bet in roulette results in an expected value of -26c and the $5 bet in craps
results in an expected value of -7c. Because you are better off losing 7c instead of
losing 26¢, the craps game is better in the long run, even though the roulette game
provides an opportunity for a larger payoff.
Rationale for Formulas 1 through 4
Instead of blindly accepting and using formulas, it is much better to have some understanding of why they work. When computing the mean from a frequency distribution, f represents class frequency and N represents population size. In the expression
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Discrete Probability Distributions
below, we rewrite the formula for the mean of a frequency table so that it applies to a
population. In the fraction f>N, the value of f is the frequency with which the value x
occurs and N is the population size, so f>N is the probability for the value of x. When
we replace f>N with P (x), we make the transition from relative frequency based on
a limited number of observations to probability based on infinitely many trials. This
result shows why Formula 1 is as given earlier in this section.
m =
Σ( f # x)
f
f#x
= ac
d = a c x # d = Σ 3 x # P (x)4
N
N
N
Similar reasoning enables us to take the variance formula and apply it
to a random variable for a probability distribution; the result is Formula 2.
Formula 3 is a shortcut version that will always produce the same result as
Formula 2. Although Formula 3 is usually easier to work with, Formula 2 is easier to
understand directly. Based on Formula 2, we can express the standard deviation as
s = 2Σ 3 (x - m)2 # P (x) 4
or as the equivalent form given in Formula 4.
2 Basic Skills and Concepts
Statistical Literacy and Critical Thinking
1. Random Variable Table 7 lists probabilities for the corresponding numbers of girls
in three births. What is the random variable, what are its possible values, and are its values
numerical?
2. Discrete or Continuous? Is the random variable given
in Table 7 discrete or continuous? Explain.
Table 7 Number of
Girls in Three Births
3. Probability Distribution Does Table 7 describe a
Number of
Girls x
P(x)
0
0.125
probability distribution? Show how the requirements are
satisfied or are not satisfied.
1
0.375
4. Unusual For 200 births, the probability of exactly 90 girls
2
0.375
is 0.0208 and the probability of 90 or fewer girls is 0.089.
3
0.125
a. Is exactly 90 girls in 200 births unlikely?
b. Among 200 births, is 90 girls an unusually low number of girls?
Identifying Discrete and Continuous Random Variables. In Exercises 5 and 6,
identify the given values as a discrete random variable, continuous random variable, or
not a random variable.
5. a. Exact weights of quarters now in circulation in the United States
b. Numbers of tosses of quarters required to get heads
c. Responses to the survey question “Did you smoke at least one cigarette in the last week?”
d. Numbers of spins of roulette wheels required to get the number 7
271
Discrete Probability Distributions
e. Exact foot lengths of humans
f. Shoe sizes (such as 8 or 8½) of humans
6. a. Eye colors of humans on commercial aircraft flights
b. Weights of humans on commercial aircraft flights
c. Numbers of passengers on commercial aircraft flights
d. Numbers of randomly generated digits before getting the digit 3
e. Political party affiliations of adults in the United States
f. Exact costs of presidential campaigns
In Exercises 7–14, determine whether a
probability distribution is given. If a probability distribution is given, find its mean
and standard deviation. If a probability distribution is not given, identify the requirements that are not satisfied.
Identifying Probability Distributions.
7. Genetic Disorder Four males with an X-linked genetic disorder
have one child each. The random variable x is the number of children among the four who inherit the X-linked genetic disorder.
8. Male Color Blindness When conducting research on color
blindness in males, a researcher forms random groups with five
males in each group. The random variable x is the number of males
in the group who have a form of color blindness (based on data from
the National Institutes of Health).
9. Pickup Line Ted is not particularly creative. He uses this pickup
line: “If I could rearrange the alphabet, I’d put U and I together.”
The random variable x is the number of girls Ted approaches before
encountering one who reacts positively.
10. Fun Ways to Flirt In a Microsoft Instant Messaging survey,
respondents were asked to choose the most fun way to flirt, and the
accompanying table is based on the results.
11. Fun Ways to Flirt A sociologist randomly selects single adults
for different groups of four, and the random variable x is the number in the group who say that the most fun way to flirt is in person
(based on a Microsoft Instant Messaging survey).
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x
0
1
2
3
4
P (x)
0.0625
0.2500
0.3750
0.2500
0.0625
x
0
1
2
3
4
5
P (x)
0.659
0.287
0.050
0.004
0.001
0+
x
1
2
3
4
5
P(x)
0.001
0.020
0.105
0.233
0.242
E-mail
In person
Instant
message
Text
message
x
0
1
2
3
4
P(x)
0.06
0.55
0.24
0.15
P (x)
0.041
0.200
0.367
0.299
0.092
Discrete Probability Distributions
12. Happiness Groups of people aged 15–65 are randomly selected
and arranged in groups of six. The random variable x is the number in the group who say that their family and/or partner contribute
most to their happiness (based on a Coca-Cola survey).
x
0
1
2
3
4
5
6
P (x)
0+
0.003
0.025
0.111
0.279
0.373
0.208
13. Happiness In a survey sponsored by Coca-Cola, subjects aged
15–65 were asked what contributes most to their happiness, and the
table is based on their responses.
Family/
partner
Friends
Work/studies
Leisure
Music
Sports
14. Casino Games When betting on the pass line in the dice game
of craps at the Mohegan Sun casino in Connecticut, the table lists
the probabilities for the number of bets that must be placed in order
to have a win.
In Exercises 15–18, refer to the accompanying table,
which describes results from groups of 10 births from 10 different
sets of parents. The random variable x represents the number of
girls among 10 children.
Genetics.
15. Mean and Standard Deviation Find the mean and standard
deviation for the numbers of girls in 10 births.
16. Range Rule of Thumb for Unusual Events Use the range
rule of thumb to identify a range of values containing the usual
numbers of girls in 10 births. Based on the result, is 1 girl in 10 births
an unusually low number of girls? Explain.
17. Using Probabilities for Unusual Events
x
1
2
3
4
5
Number of
Girls x
0
1
2
3
4
5
6
7
8
9
10
P (x)
0.77
0.15
0.08
0.08
0.06
0.04
P (x)
0.493
0.250
0.127
0.064
0.033
P(x)
0.001
0.010
0.044
0.117
0.205
0.246
0.205
0.117
0.044
0.010
0.001
a. Find the probability of getting exactly 8 girls in 10 births.
b. Find the probability of getting 8 or more girls in 10 births.
c. Which probability is relevant for determining whether 8 is an unusually high number of
girls in 10 births: the result from part (a) or part (b)?
d. Is 8 an unusually high number of girls in 10 births? Why or why not?
18. Using Probabilities for Unusual Events
a. Find the probability of getting exactly 1 girl in 10 births.
b. Find the probability of getting 1 or fewer girls in 10 births.
c. Which probability is relevant for determining whether 1 is an unusually low number of
girls in 10 births: the result from part (a) or part (b)?
d. Is 1 an unusually low number of girls in 10 births? Why or why not?
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