STAT 200: Introduction to Statistics
Homework #4 Solutions
1. (3 points): The commuter trains on the Red Line for the Regional Transit Authority (RTA) in Cleveland,
OH, have a waiting time during peak rush hour period
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STAT 200: Introduction to Statistics
Homework #4 Solutions
1. (3 points): The commuter trains on the Red Line for the Regional Transit Authority (RTA) in Cleveland,
OH, have a waiting time during peak rush hour periods of eight minutes ("2012 annual report," 2012).
a.) State the random variable.
The random variable is given by: x = waiting times for the RTA
b.) Find the height of this uniform distribution.
It will be beneficial to draw a diagram of this:
We know that the sum of all probabilities for a probability distribution must be 1. Since the length of the
“rectangle” defined by the blue lines above is 8 – 0 = 8, and its area is 1, using the definition of area = length
* width (or in this case, area = length * height), we have: 1 = 8 * h h = 1/8 = 0.125.
c.) Find the probability of waiting between four and five minutes.
This concept is given by the diagram below. The total area of the blue rectangle is 1. We want to find the
area of the red rectangle, which is given by length * height. We found the height in part B; it was 1/12 =
0.125. The length is given by: 5 – 4 = 1.
Thus, the area of the red rectangle is length * height = 1 * 1/12 = 1 * 0.125 = 0.125.
-2 -1 0 1 2 3 4 5 6 7 8 9 10
-2 -1 0 1 2 3 4 5 6 7 8 9 10
Height =
1/8 = 0.125
Height =
1/8 = 0.125
Page 2 of 28
d.) Find the probability of waiting between three and eight minutes.
This concept is given by the diagram below. The total area of the blue rectangle is 1. We want to find the
area of the red rectangle, which is given by length * height. We found the height in part B; it was 1/12 =
0.0125. The length is given by: 8 – 3 = 5.
Thus, the area of the red rectangle is length * height = 5 * 1/12 = 5/12=0.625.
e.) Find the probability of waiting five minutes exactly
This concept is given by the diagram below. The total area of the blue rectangle is 1. We want to find the
area of the red rectangle, which is given by length * height. We found the height in part B; it was 1/12 =
0.125. The length is given by: 5 – 5 = 0.
Thus, the area of the red rectangle is length * height = 0 * 0.125 = 0.
-2 -1 0 1 2 3 4 5 6 7 8 9 10
-2 -1 0 1 2 3 4 5 6 7 8 9 10
Height =
1/8 = 0.125
Height =
1/8 = 0.125
Page 3 of 28
2. (3 points): Find the z-score corresponding to the given area. Remember, z is distributed as the standard
normal distribution with mean of µ = 0 and standard deviation σ = 1.
a.) The area to the left of z is 15%.
A graph may aid in understanding of this problem:
NOTE: These values can be found using a standard normal distribution table or Microsoft Excel.
We will look at Excel first… The equation is: NORM.S.INV which
Returns the inverse of the standard normal cumulative distribution. The distribution has a mean of zero
and a standard deviation of one.
Syntax: NORM.S.INV(probability)
The NORM.S.INV function syntax has the following arguments:
• Probability: Required. A probability corresponding to the normal distribution.
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