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Instructions: Use the space provided to enter your answers. Be sure to use the Microsoft
Equation Editor to show answers and calculations.

1) Suppose x is a random variable best described by a uniform probability
distribution with c=10 and d=30 Show calculations for each computation.

a) Find f(x)

1
{
b) Find the mean and standard
deviation.
20
, 10 < 𝑥 < 30
0
𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
10
Mean =20, sd =
√3
If x has a uniform distribution on the interval (c,d)
Then mean =
𝑐+𝑑
2
𝑑−𝑐
Sd =
√12
If c=10 and d=30
Then mean =
𝑑−𝑐
Sd =
√12
=
𝑐+𝑑
2
30−10
√12
=
10+30
2
20
= 2√3 =
=
40
2
= 20
10
√3

2) Find the value of of the standard normal random variable for the following
probabilities. Show calculations for each computation.

a) P(z ≤ z_0 )=.95
z0 = 1.645

b) P(-2 < z < z_0)=.9710
z0=2.512
a) Using normal table P(z≤1.645)=0.95
b) P(-2 < z < z0) =P(z<z0)-P(z<-2)=P(z<z0)-0.023=0.9710
P(z<z0)=0.9710+0.023=0.994
z0=2.512 (using normal table)

3) Suppose is a normally distributed random variable with μ=11and σ=2. Find each
of the following. Show calculations for each computation.

a) P(10≤x≤12)
0.383

b) P(x≥13.24)
0.131
Z=(X- μ)/ σ =(X-11)/2has a standard normal distribution
a) P(10≤x≤12)=P((10-11)/2≤Z≤(12-11)/2)=
P(-1/2≤Z≤1/2)=P(z≤0.5)- P(z≤0.5)=0.383
b) P(x≥13.24)=P(Z≥(13.24-11)/2)=P(Z≥1.12)=1-P(Z≤1.12)=
1-0.869=0.131

4) Refer to the Chance (Winter 2001) study of students who paid a private tutor to
help them improve their SAT scores, presented in Exercise 2.101 (p.74). The table
summarizing the changes in both the SAT-Mathematics and SAT-Verbal scores for
these students is reproduced here. Assume that both distributions of SAT score
changes are approximately normal.
SAT-Math SAT-Verbal
19
7
Mean change in score
65
49
Standard deviation of changes in score.
Compute the probability that a student increases his or her score on the SAT-Math test by at
least 50 points? Show calculations.
X= increase of the score
X has a normal distribution with Mean = 19 and sd=65
Z= (X-19)/65 has a standard normal distribution
P(X>50)=P(Z>(50-19)/65)=P(Z>0.477)=0.317
Answer: 0.317