Download Midterm Sample Test

Math 170/171
A sample of the midterm exam questions
1. Classify the following variables as qualitative or quantitative. Then give the level of
the measurement of it.
The color of the cars parked in parking lot A at Richland.
2. The mean of a distribution is 50 with a standard deviation of 5. Use Chebyshev’s
theorem to answer the following questions.
At least what percent of the data lie between 35 and 65?
3. Students in my classes are either full-time or part-time. They either own a house or
rent one. Suppose, this semester, I have 100 students. Out of 100 students 40 of them are
working full-time, 30 of them own a house, and 25 of them are working full-time and
own a house. Make a nice table, which summarizes the above information, and then
answer the following questions. If I randomly select one of my student this semester,
what is the probability that the student (4 points for the table.)
a. Owns a house;
b. Are events owning a house and being full-time independent? Explain mathematically.
4. A box contains 10 $1 bills, 20 $2 bills, 5 $5 bills, 3 $10 bills, and 1 $100 bill. A
person is charged $20 to select one bill randomly. Find the expectation. Is this a fair
game? Explain.
Can you make up a friendly game and explain why you think it is a fair game?
5. There are 12 students in my online class. Seven are male and 5 are female. We
randomly select 5 students for a committee. What is the probability that:
At least one female?
6. A quality control manager says that 1% of the parts they manufacture is defective. A
customer randomly selects 300 of these parts. Find the probability that at least 2 items
are defective.
Use a Poisson distribution. Hint: The given distribution is a binomial one. Note that
μ = np will give you the mean.
7. Given that a normal distribution has μ = 10 cm and σ 2 = 4 cm2 . Find p85 .
8. The weight of ripe watermelons grown at Mr. Smith’s farm are normally distributed
with a standard deviation of 3 pounds. Find the mean weight of Mr. smith’s watermelons
if only 6% weigh less than 15 pounds.
9. Ten percent of the population is left-handed. In a class of 100 students, what is the
probability that 12 are left-handed? Use a normal distribution to approximate it.
10. Assume that the salaries of elementary teachers in the United States are normally
distributed with a mean of $31000 and standard deviation of $3000. What is the cutoff
salary for teachers in the top 10%?
Answers:
1. Qualitative, Nominal L. of M
2.
65 − 50
k=
=3
5
1
1 8
1 − 2 = 1 − 2 = ⇒ ≈ 89%
9
k
3
3b. If p ( O F ) = p ( O ) , then events O & F are independent.
p (O F ) =
25 5
3
= ≠ p (O ) =
40 8
10
∴ Events O & F are dependent.
4.
x
p ( x)
x ⋅ p ( x)
1
10
10
39
2
20
39
5
5
39
10
3
39
100 1
39
1
Σ
39
40
39
25
39
30
39
100
39
205
39
E ( x) =
205
− 20 = −14.74
39
Since E ( x ) ≠ 0 , then this is not a friendly game.
5.
p ( x ≥ 1) = 1 − p ( x = 0 ) = 1 −
7 6 5 4 3
7
⋅ ⋅ ⋅ ⋅ = 1−
12 11 10 9 8
12 ⋅11 ⋅ 2
6.
μ = n ⋅ p = (.01)( 300 ) = 3
μ x e− μ
p ( x) =
, p ( 0 ) = e −3 , p (1) = 3e −3
x!
p ( x ≥ 2 ) = 1 − ( e−3 + 3e−3 ) = 1 − 4e−3 = .801
7. z =
8.
z=
x−μ
σ
x−μ
σ
−1.555 =
⇒ x = μ + zσ = 10 + ( 2 )(1.04 ) = 12.08 cm
, − z.06 = −1.555
15 − μ
⇒ μ = 15 − 3 ( −1.555 ) = 19.7 lb
3
9.
p = .1
p ( x = 12 ) = p (11.5 p x p 12.5 )
12.5 − 10 ⎞
⎛ 11.5 − 10
= p⎜
pzp
⎟
3
3
⎝
⎠
= p (.5 p z p .83) = p ( z p .83) − p ( z p .5 )
n = 100
μ = np = 10
σ = npq =
(.1)(.9 )(100 )
= .7967 − .6915 = .1052
σ =3
10.
zC = 1.28
$34,840
x = μ + zσ = 31000 + 1.28 ( 3000 ) = 34,840