Download 60. Refer to the Real Estate data, which report information on homes

60. Refer to the Real Estate data, which report information on homes sold in the Phoenix, Arizona, area last
year. a. Create a probability distribution for the number of bedrooms. Compute the mean and the standard
deviation of this distribution. b. Create a probability distribution for the number of bathrooms. Compute the
mean and the standard deviation of this distribution.
(a)
Number of Beds(x)
2
3
4
5
6
7
8
Frequency (f)
24
26
26
11
14
2
2
P(x)
0.2286
0.2476
0.2476
0.1048
0.1333
0.0190
0.0190
x * P(x)
0.4571
0.7429
0.9905
0.5238
0.8000
0.1333
0.1524
x^2 * P(x)
0.9143
2.2286
3.9619
2.6190
4.8000
0.9333
1.2190
Sums =
105
1
3.8000
16.6762
Mean = ∑(x * P(x)) = 3.8
Variance = [∑(x^2 * P(x)] - Mean^2 = 16.6762 - 3.8^2 = 2.2362
Standard deviation = √Variance = √2.2362 = 1.495
(b)
Number of Baths
(x)
1.5
2
2.5
3
Frequency
(f)
16
65
15
9
P(x)
0.1524
0.6190
0.1429
0.0857
x * P(x)
0.2286
1.2381
0.3571
0.2571
x^2 * P(x)
0.3429
2.4762
0.8929
0.7714
Sums =
105
1
2.0810
4.4833
Mean = ∑(x * P(x)) = 2.08
Variance = [∑(x^2 * P(x)] - Mean^2 = 4.4833 - 2.08^2 = 0.1569
Standard deviation = √Variance = √0.1569 = 0.396
54. Refer to the Real Estate data, which report information on homes sold in the Phoenix, Arizona, area during
the last year. a. The mean selling price (in $ thousands) of the homes was computed earlier to be $221.10, with
a standard deviation of $47.11. Use the normal distribution to estimate the percent of homes selling for more
than $280.0. Compare this to the actual results. Does the normal distribution yield a good approximation of the
actual results? b. The mean distance from the center of the city is 14.629 miles with a standard deviation of
4.874 miles. Use the normal distribution to estimate the number of homes 18 or more miles but less than 22
miles from the center of the city. Compare this to the actual results. Does the normal distribution yield a good
approximation of the actual results?
(a)
μ = 221.10, σ = 47.11, x = 280
z = (x - μ)/σ = (280 - 221.10)/47.11 = 1.2503
P(x > $280) = P(z > 1.2503) = 0.1056
Number of homes selling for more than $280 = 105 * 0.1056 = 11.09 (say 11)
Actual number of homes selling for more than $280 = 14
The normal distribution does not give a good approximation.
(b)
μ = 14.629, σ = 4.874, x1 = 18, x2 = 22
z = (x - μ)/σ = (280 - 221.10)/47.11 = 1.2503
z1 = (x1 - μ)/σ = (18 - 14.629)/4.874 = 0.6916 and z2 = (x2 - μ)/σ = (22 - 14.629)/4.874 = 1.5123
P(18 miles ≤ x < 22 miles)= P(0.6916 ≤ z < 1.5123) = 0.1794
Number of homes 18 or more miles but less than 22 miles from the city center = 105 * 0.1794 = 18.84 (say 19)
Actual number of homes 18 or more miles but less than 22 miles from the city center = 24
The normal distribution does not give a good approximation.