Stats 7 winter 2019 (Baldi) – Midterm2 – SOLUTIONS

Problem 1 (16 points) < 12 min

A manufacturer makes a component to be used for a car engine. The manufacturing process is
such that the component diameter varies approximately Normally with a mean 9.1 millimeters (mm)
and standard deviation 1.5 mm. Show your detailed work for all questions in this problem.
a) Obtain the first quartile of the distribution of diameters.
Q1 = invNorm(.25, 9.1, 1.5) = 8.09 mm

b) The component is considered defective if its diameter is either less than 8.0 mm or greater than
10.5 mm. What is the probability that a randomly selected component is defective?
P(X < 8.0 or X > 10.5) = 1 – P(8.0 < X < 10.5) = 1 – normalcdf(8.0, 10.5, 9.1, 1.5) = 0.4070
P(X < 8.0 or X > 10.5) = normalcdf(-1E99, 8.0, 9.1, 1.5) + normalcdf(10.5, 1E99, 9.1, 1.5)
= 0.2317 + 0.1753 = 0.4070 (or 40.7%)
c) How large are the 5% largest diameters of all components produced?
x = invNorm(0.95, 9.1, 1.5) = 11.57  They have diameters of 11.57 mm and greater.

d) What is the probability that a random sample of 25 components would have a mean diameter
greater than 9.5 mm? P(Xbar > 9.5) = normalcdf(9.5, 1E99, 9.1, 1.5/√25) = 0.0912 (or 9.12%)
[This question requires using the sampling distribution of xbar]

Problem 2 (6 points) < 3 min

Chicago is an old city with lead water pipes. The U.S. Environmental Protection Agency (EPA)
evaluated water lead levels in a random sample of 32 homes in Chicago. The findings showed a
mean water lead level of 4.76 mcg/l (95% confidence interval 3.85 to 5.67 mcg/l).
a) Interpret this confidence interval in context.
We are 95% confident that the mean water lead level in all the homes in Chicago is a value
between 3.85 and 5.67 mcg/l.
b) The value of the margin of error for this interval is 0.91 mcg/l.

Problem 3 (12 points) < 12 min

The CDC reports that there were 3,855,500 births in the United States for the year 2017. The
report breaks down the births by age of the mother and by type of delivery (vaginal, cesarean, or
unknown/not stated).

Type of delivery All delivery

Mother's age Vaginal Cesarean Not Stated types
Under 20 156,727 39,459 108 196,294
20–24 566,567 197,719 494 764,780
25–29 784,533 338,366 678 1,123,577
30–34 717,581 373,825 511 1,091,917
35–39 331,394 223,119 283 554,796
40–54 64,208 59,851 77 124,136
All ages 2,621,010 1,232,339 2,151 3,855,500

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a) Obtain the following probability values. Show your work.
P(cesarean and 20-24) = 197,719 / 3,855,500 = 0.0513 (5.13%)
P(cesarean | 20-24) = 197,719 / 764,780 = 0.2585 (25.85%)
P(20-24 | cesarean) = 197,719 /1,232,339 = 0.1604 (16.04%)

b) Fill in the blanks:

In 2017 in the United Stated, 31.96 percent of all births were delivered via cesarean, but 20.10
percent of births to young mothers under the age 20 were delivered via cesarean. That year, 5.09
percent of all births were births to young mothers under the age of 20.

Problem 4 (12 points) < 12 min

Earlier this month, the Maryland State’s Attorney announced that she will no longer prosecute
marijuana possession cases on behalf of the state. Her statement said, "Prosecuting these cases
have no public safety value, disproportionately impacts communities of color and erodes public
trust, and is a costly and counterproductive use of limited resources." Among the arguments she
listed was this: "Though white and Black residents use marijuana at roughly the same rates,
marijuana laws have been and continue to be disproportionately enforced against people of color."

Part 1: The state of Maryland’s arrest rate for marijuana possession in 2010 was the fourth highest
in the nation. Public records for that year show that police arrested one out of every 250 Maryland
residents for possession of marijuana. In addition, while Black people only comprised 30% of the
State's population in 2010, 58% of those arrested for marijuana possession were Black.
Use the following labels for events describing Maryland residents in the year 2010.
M-AMP: Maryland resident was arrested for marijuana possession
M-B: Maryland resident was a Black person
a) What is the probability that a Maryland resident was arrested for marijuana possession in 2010?
P(M-AMP) = 1/250 = 0.004 (0.4%)

b) Use probability notation to express the cited percent values.

30% = P(M-B) ; 58% = P(M-B | M-AMP)
c) Which of the values cited should be compared to determine whether race and arrests for
marijuana possession in the state of Maryland in 2010 are independent?
30% and 58% [P(M-B) and P(M-B | M-AMP)]

Part 2: In 2014, the State of Maryland decriminalized possession of small amounts of marijuana to
just a civil infraction [similar to a parking ticket]. The State’s Attorney points out that, “in 2017,
police officers in the city of Baltimore [in Maryland] issued 431 citations for marijuana possession,
where 410 (95%) were issued to Black people.” However, “the city of Baltimore is home to 622,454
residents, 62% of which are Black.”
Use the following notation for events describing Baltimore residents in the year 2017.
B-CMP: Baltimore resident received a citation for marijuana possession
B-B: Baltimore resident was a Black person
Using probability notation “P(…),” describe in one equation the argument used by the Maryland
State’s Attorney for Baltimore city as evidence that marijuana laws are disproportionately enforced
against people of color in the city of Baltimore.
P(B-B | B-CMP)] > P(B-B) [95% > 62%]

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Problem 5 (4 points) < 3 min
Firearms are not allowed in carry-on bags on commercial airplanes. The Transportation Security
Administration (TSA) published earlier this month its official report for the year 2018. According to
the report, TSA officers discovered 4,239 firearms in carry-on bags at airport checkpoints across
the country last year, representing an average daily discovery of 11.6 firearms. Most of the firearms
discovered in carry-on bags were loaded [ready to fire].
a) The cited value 11.6 is: A) biased B) an estimate C) a parameter D) confident E) a statistic
b) In describing daily checks of carry-on bags by the
TSA at US airports for the year 2018, the cited value 𝒔 - 𝒏 - 𝝁 - 𝒑 - 𝒙 - 𝝈 - 𝒎 - 𝒑 - 𝑪
11.6 corresponds to which symbol?

Problem 6 (EXTRA CREDIT)

A large survey of Americans was conducted in 2017 to examine religion and culture issues in the
United States. The resulting report included the following graphic:

We can use the results of this survey to create an approximate probability model for the American
population. For simplicity, use the following notation to describe events in that sample space:
LDAM: person says there is a lot of discrimination against Muslims
LDAC: person says there is a lot of discrimination against Christians
a) Express the last value 23% using probability notation: 23% = P(LDAC | Unaffiliated)
b) The last two values, 77% and 23%, represent probabilities for events that are
A) complementary and disjoint C) disjoint but not complementary
B) complementary but not disjoint D) not disjoint and not complementary
c) Which is an appropriate interpretation?
A) Americans in general say that there is a lot more discrimination against Muslims than there
is against Christians, but White evangelical Protestants say that there is a lot more
discrimination against Christians than there is against Muslims.
B) Americans in general are more likely to say that there is a lot of discrimination
against Muslims than to say that there is a lot of discrimination against Christians,
but White evangelical Protestants are more likely to say that there is a lot of
discrimination against Christians than to say that there is a lot of discrimination
against Muslims.
C) Americans in general say that Muslims experience a lot more discrimination than Christians.
In contrast, White evangelical Protestants say that both Christians and Muslims experience
a lot of discrimination but that Christians tend to experience a bit more discrimination than
Muslims.
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