Concept explainers
a.
Compute the
a.
Answer to Problem 117CE
The mean is approximately $1,100K.
Explanation of Solution
Calculation:
It is given that the distribution of beach condominium prices (in $ thousands) follows triangular distribution with lower limit 500,
Triangular distribution:
A continuous random variable X is said to follow triangular distribution if the probability density
f(x)={2(x−a)(b−a)(c−a) for a≤x≤b2(c−x)(c−a)(c−b) for b≤x≤c ,
where a, b and c are the lower limit, mode and upper limit, respectively.
Mean of a Triangular distribution is defined as,
μ=a+b+c3.
It is given that, a=500, b=700 and c=2,100.
Thus, the mean of the Triangular distribution is,
μ=500+700+2,1003=3,3003=1,100_.
Therefore, the mean is $1,100K.
b.
Compute the standard deviation.
b.
Answer to Problem 117CE
The standard deviation is $355.90K.
Explanation of Solution
Calculation:
The standard deviation of a Triangular distribution is defined as,
σ=√a2+b2+c2−ab−ac−bc18.
It is given that, a=500, b=700 and c=2,100.
Thus, the standard deviation of the Triangular distribution is,
σ=√a2+b2+c2−ab−ac−bc18=√5002+7002+2,1002−(500)(700)−(500)(2,100)−(700)(2,100)18=√250,000+490,000+4,410,000−350,000−1,050,000−1,470,00018=√2,280,00018=√126,666.67≈355.90_.
Therefore, the standard deviation of the Triangular distribution is $355.90K.
c.
Compute the probability that a condo price will be greater than $750K.
c.
Answer to Problem 117CE
The probability that condo price will be greater than $750K is 0.8136.
Explanation of Solution
Calculation:
The cumulative distribution function (CDF) of a Triangular distribution is defined as,
P(X≤x)={(x−a)2(b−a)(c−a) for a≤x≤b1−(c−x)2(c−a)(c−b) for b≤x≤c .
Assume that the random variable X denotes the beach condominium prices.
It is given that, a=500, b=700, c=2,100 and x=750.
Hence, the CDF of the Triangular distribution will be,
P(X≤x)=1−(c−x)2(c−a)(c−b) for b≤x≤c.
Thus, the probability that condo price will be greater than $750K is,
P(X>750)=(2,100−750)2(2,100−500)(2,100−700)=1,3502(1,600)(1,400)=1,822,5002,240,000=0.8136_.
Therefore, the probability that condo price will be greater than $750K is 0.8136.
d.
Draw the distribution and shade the area for the
d.
Answer to Problem 117CE
The shaded sketch is obtained as:
Explanation of Solution
Calculation:
It is given that, a=500, b=700 and c=2,100.
Graphical Procedure:
Step by step procedure to obtain the sketch is given below:
- Draw a right skewed triangle with base from the lower limit 500 to upper limit 2,100 and with the height at the mode of 700.
- Shade the right side area from the mode of 750.
The shaded sketch is obtained as:
The shaded area in the sketch is the event in part (c).
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Chapter 7 Solutions
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