I. Over the past 300 days, DiCarlo has experienced 54 days with no automobiles s
ID: 3353328 • Letter: I
Question
I. Over the past 300 days, DiCarlo has experienced 54 days with no automobiles sold, 117 days with 1 automobile sold, 72 days with 2 automobiles sold, 42 days with 3 automobiles sold, 12 days with 4 automobiles sold, and 3 days with 5 automobiles sold. Suppose we consider the experiment of observing a day of operations at DiCarlo Motors and define the random variable of interest as x the number of automobiles sold during a day. a) Use the relative frequency method to develop a probability distribution for the number of cars sold per day at DiCarlo Motors in Saratoga, New York. b) The expected value for the number of automobiles sold during a day c) The variance for the number of automobiles sold during a dayExplanation / Answer
Given x = number of automobiles sold during a day, the given frequency and the related requisite calculations are shown in the following table.
i
Number of automobiles sold during the day
(xi)
Number of days
(Frequency)
(fi)
Relative
Frequency
Ri = p(xi)
= fi/300
xi.p(xi)
xi2.p(xi)
1
0
54
0.18
0.00
0.00
2
1
117
0.39
0.39
0.39
3
2
72
0.24
0.48
0.96
4
3
42
0.14
0.42
1.26
5
4
12
0.04
0.16
0.64
6
5
3
0.01
0.05
0.25
Total
300
1.00
1.50
3.50
Part (a)
Probability Distribution
Number of automobiles sold during the day
(xi)
Probability
0
0.18
1
0.39
2
0.24
3
0.14
4
0.04
5
0.01
Total
1.00
ANSWER
Part (b)
Expected number of cars sold per day = E(X) = [i = 1, 6]{xi.p(xi)} = 1.50 ANSWER [refer to column 5 of table at the top]
Part (c)
Variance of the number of cars sold per day = V(X)
= E(X2) – { E(X)}2
= [i = 1, 6]{xi2.p(xi)} – 1.52 [refer to the answer of Part (b)]
= 3.5 – 2.25 [refer to column 6 of table at the top]
= 1.25 ANSWER
i
Number of automobiles sold during the day
(xi)
Number of days
(Frequency)
(fi)
Relative
Frequency
Ri = p(xi)
= fi/300
xi.p(xi)
xi2.p(xi)
1
0
54
0.18
0.00
0.00
2
1
117
0.39
0.39
0.39
3
2
72
0.24
0.48
0.96
4
3
42
0.14
0.42
1.26
5
4
12
0.04
0.16
0.64
6
5
3
0.01
0.05
0.25
Total
300
1.00
1.50
3.50