Download Chapter 3: Describing Relationships (first spread)

Statistics: Chapter 5.1, 5.2, 5.3 ATE: Probability: What
Are the Chances?
Alternate Activities and Examples
[Page 203]
Alternate Example: AP Statistics Scores
Randomly select a student who took the 2010 AP Statistics exam and record the student’s score.
Here is the probability model:
Score
1
2
3
4
5
Probability 0.223 0.183 0.235 0.224 0.125
Problem:
(a) Show that this is a legitimate probability model.
(b) Find the probability that the chosen student scored 3 or better.
Solution:
[Page 220]
Alternate Example: Who Owns a Home?
What is the relationship between educational achievement and home ownership? A random
sample of 500 people who participated in the 2000 census was chosen. Each member of the
sample was identified as a high school graduate (or not) and as a home owner (or not). The twoway table displays the data.
High
School
Graduate
Homeowner
Not a Homeowner
Total
221
89
310
Not a
High
School
Graduate
119
71
190
Total
340
160
500
Problem: Suppose we choose a member of the sample at random. Find the probability that the
member
(a) is a high school graduate
(b) is a high school graduate and owns a home
(c) is a high school graduate or owns a home
Solution: We will define event A as being a high school graduate and event B as being a
homeowner.
The Practice of Statistics for AP*, 4/e
© BFW Publishers 2011
[Page 220]
Alternate Example: Who Owns a Home?
Here is the two-way table summarizing the relationship between educational status and home
ownership from the previous example:
High
School
Graduate
Homeowner
Not a Homeowner
Total
221
89
310
Not a
High
School
Graduate
119
71
190
Total
340
160
500
The four distinct regions in the Venn diagram below correspond to the four non-total cells in the
two-way table as follows: (fill in the counts below and in the Venn diagram)
Region in Venn Diagram
In the intersection of two circles
Inside circle A, outside circle B
Inside circle B, outside circle A
Outside both circles
A
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In Words
HS grad and owns home
HS grad and doesn’t own home
Not HS grad but owns home
Not HS grad and doesn’t own home
In Symbols
A B
A  Bc
Ac  B
Ac  Bc
B
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Count
[Page 220]
Alternate Example: Phone Usage
According to the National Center for Health Statistics,
(http://www.cdc.gov/nchs/data/nhis/earlyrelease/wireless200905_tables.htm#T1), in December
2008, 78% of US households had a traditional landline telephone, 80% of households had cell
phones, and 60% had both. Suppose we randomly selected a household in December 2008.
Problem:
(a) Make a two-way table that displays the sample space of this chance process.
(b) Construct a Venn diagram to represent the outcomes of this chance process.
(c) Find the probability that the household has at least one of the two types of phones.
(d) Find the probability the household has a cell phone only.
Solution:
We will define events A: has a landline and B: has a cell phone.
(a)
Cell Phone No Cell Phone Total
Landline
No Landline
Total
(b)
A: Landline
B: Cell phone
(c)
(d)
The Practice of Statistics for AP*, 4/e
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5.3 Alternate Examples
[Page 226]
Alternate Example: Who Owns a Home?
High
School
Graduate
Homeowner
Not a Homeowner
Total
221
89
310
Not a
High
School
Graduate
119
71
190
Total
340
160
500
1. If we know that a person owns a home, what is the probability that the person is a high
school graduate?
2. If we know that a person is a high school graduate, what is the probability that the person
owns a home?
[Page 228]
Alternate Example: Who Owns a Home?
The events of interest in this scenario were A: is a high school graduate and B: owns a home. We
already learned that P(B) = 340/500 = 68% and that P(B | A) = 221/310 = 71.2%. That is, we
know that a randomly selected member of the sample has a 68% probability of owning a home.
However, if we know that the randomly selected member is a high school graduate, the
probability of owning a home increases to 71.2%.
The Practice of Statistics for AP*, 4/e
© BFW Publishers 2011
[Page 228]
Alternate Example: Allergies
Is there a relationship between gender and having allergies? To find out, we used the random
sampler at the United States Census at School website (www.amstat.org/censusatschool) to
randomly select 40 US high school students who completed a survey. The two-way table shows
the gender of each student and whether the student has allergies.
Female Male Total
Allergies
10
8
18
No Allergies
13
9
22
Total
23
17
40
Problem: Are the events “female” and “allergies” independent? Justify your answer.
Solution:
[Page 230]
Alternate Example: Picking Two Sneezers
In the previous alternate example, we used a two-way table that classified 40 students according
to their gender and whether they had allergies. Here is the table again.
Female Male Total
Allergies
10
8
18
No Allergies
13
9
22
Total
23
17
40
Problem: Suppose we chose 2 students at random.
(a) Draw a tree diagram that shows the sample space for this chance process.
(b) Find the probability that both students suffer from allergies.
Solution:
The Practice of Statistics for AP*, 4/e
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[Page 230]
Alternate Example: Playing in the NCAA
About 55% of high school students participate in a school athletic team at some level and about
5% of these athletes go on to play on a college team in the NCAA.
(http://www.washingtonpost.com/wp-dyn/content/article/2009/09/23/AR2009092301947.html,
http://www.collegesportsscholarships.com/percentage-high-school-athletes-ncaa-college.htm)
Problem: What percent of high school students play a sport in high school and go on to play a
sport in the NCAA?
Solution: We know P(high school sport) = 0.55 and P(NCAA sport | high school sport) = 0.05,
so
[Page 235] Ex 5.28
Alternate Example: Perfect Games
In baseball, a perfect game is when a pitcher doesn’t allow any hitters to reach base in all nine
innings. Historically, pitchers throw a perfect inning—an inning where no hitters reach base—
about 40% of the time (http://www.baseballprospectus.com/article.php?articleid=11110). So, to
throw a perfect game, a pitcher needs to have nine perfect innings in a row.
Problem: What is the probability that a pitcher throws nine perfect innings in a row, assuming
the pitcher’s performance in an inning is independent of his performance in other innings?
Solution
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[Page 235] Ex 5.29
Alternate Example: First Trimester Screen
The First Trimester Screen is a non-invasive test given during the first trimester of pregnancy to
determine if there are specific chromosomal abnormalities in the fetus. According to a study
published in the New England Journal of Medicine in November 2005
(http://www.americanpregnancy.org/prenataltesting/firstscreen.html), approximately 5% of
normal pregnancies will receive a positive result. Among 100 women with normal pregnancies,
what is the probability that there will be at least one false positive?
[Page 229]
Alternate Example: Phone Usage
In an alternate example in section 5.2, we classified US households according to the types of
phones they used.
Cell Phone No Cell Phone Total
Landline
0.60
0.18
0.78
No Landline
0.20
0.02
0.22
Total
0.80
0.20
1.00
Problem: What is the probability that a randomly selected household with a landline also has a
cell phone?
Solution:
The Practice of Statistics for AP*, 4/e
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[Page ]
Alternate Example: False Positives and Drug Testing
Many employers require prospective employees to take a drug test. A positive result on this test
indicates that the prospective employee uses illegal drugs. However, not all people who test
positive actually use drugs. Suppose that 4% of prospective employees use drugs, the false
positive rate is 5% and the false negative rate is 10%.
(http://www.cbsnews.com/stories/2010/06/01/health/webmd/main6537635.shtml)
Problem: What percent of people who test positive actually use illegal drugs?
Solution: The tree diagram summarizes this situation.
.
The Practice of Statistics for AP*, 4/e
© BFW Publishers 2011