1. Refer to the Real Estate data, which report information on the homes sold in
ID: 3265328 • Letter: 1
Question
1. Refer to the Real Estate data, which report information on the homes sold in Goodyear, Arizona, last year.
a. At the .01 significance level, can we conclude that there is a difference in the mean selling price of homes with a pool and homes without a pool?
b. At the .01 significance level, can we conclude that there is a difference in the mean selling price of homes with an attached garage and homes without an attached garage?
c. At the .01 significance level, can we conclude that there is a difference in the mean selling price of homes in Township 1 and Township 2?
d. Find the median selling price of the homes. Divide the homes into two groups, those that sold for more than (or equal to) the median price and those that sold for less. Is there a difference in the proportion of homes with a pool for those that sold at or above the median price versus those that sold for less than the median price? Use the .01 significance level.
e. Write a summary report on your findings to parts (a), (b), (c), and (d). Address the report to all real estate agents who sell property in Goodyear.
X1 X2 X3 X4 X5 X6 X7 X8 263.1 4 2,300 0 17 5 1 2.0 182.4 4 2,100 1 19 4 0 2.0 242.1 3 2,300 1 12 3 0 2.0 213.6 2 2,200 1 16 2 0 2.5 139.9 2 2,100 1 28 1 0 1.5 245.4 2 2,100 0 12 1 1 2.0 327.2 6 2,500 1 15 3 1 2.0 271.8 2 2,100 1 9 2 1 2.5 221.1 3 2,300 0 18 1 0 1.5 266.6 4 2,400 1 13 4 1 2.0 292.4 4 2,100 1 14 3 1 2.0 209.0 2 1,700 1 8 4 1 1.5 270.8 6 2,500 1 7 4 1 2.0 246.1 4 2,100 1 18 3 1 2.0 194.4 2 2,300 1 11 3 0 2.0 281.3 3 2,100 1 16 2 1 2.0 172.7 4 2,200 0 16 3 0 2.0 207.5 5 2,300 0 21 4 0 2.5 198.9 3 2,200 0 10 4 1 2.0 209.3 6 1,900 0 15 4 1 2.0 252.3 4 2,600 1 8 4 1 2.0 192.9 4 1,900 0 14 2 1 2.5 209.3 5 2,100 1 20 5 0 1.5 345.3 8 2,600 1 9 4 1 2.0 326.3 6 2,100 1 11 5 1 3.0 173.1 2 2,200 0 21 5 1 1.5 187.0 2 1,900 1 26 5 0 2.0 257.2 2 2,100 1 9 4 1 2.0 233.0 3 2,200 1 14 3 1 1.5 180.4 2 2,000 1 11 5 0 2.0 234.0 2 1,700 1 19 3 1 2.0 207.1 2 2,000 1 11 5 1 2.0 247.7 5 2,400 1 16 2 1 2.0 166.2 3 2,000 0 16 2 1 2.0 177.1 2 1,900 1 10 5 1 2.0 182.7 4 2,000 0 14 4 0 2.5 216.0 4 2,300 1 19 2 0 2.0 312.1 6 2,600 1 7 5 1 2.5 199.8 3 2,100 1 19 3 1 2.0 273.2 5 2,200 1 16 2 1 3.0 206.0 3 2,100 0 9 3 0 1.5 232.2 3 1,900 0 16 1 1 1.5 198.3 4 2,100 0 19 1 1 1.5 205.1 3 2,000 0 20 4 0 2.0 175.6 4 2,300 0 24 4 1 2.0 307.8 3 2,400 0 21 2 1 3.0 269.2 5 2,200 1 8 5 1 3.0 224.8 3 2,200 1 17 1 1 2.5 171.6 3 2,000 0 16 4 0 2.0 216.8 3 2,200 1 15 1 1 2.0 192.6 6 2,200 0 14 1 0 2.0 236.4 5 2,200 1 20 3 1 2.0 172.4 3 2,200 1 23 3 0 2.0 251.4 3 1,900 1 12 2 1 2.0 246.0 6 2,300 1 7 3 1 3.0 147.4 6 1,700 0 12 1 0 2.0 176.0 4 2,200 1 15 1 1 2.0 228.4 3 2,300 1 17 5 1 1.5 166.5 3 1,600 0 19 3 0 2.5 189.4 4 2,200 1 24 1 1 2.0 312.1 7 2,400 1 13 3 1 3.0 289.8 6 2,000 1 21 3 1 3.0 269.9 5 2,200 0 11 4 1 2.5 154.3 2 2,000 1 13 2 0 2.0 222.1 2 2,100 1 9 5 1 2.0 209.7 5 2,200 0 13 2 1 2.0 190.9 3 2,200 0 18 3 1 2.0 254.3 4 2,500 0 15 3 1 2.0 207.5 3 2,100 0 10 2 0 2.0 209.7 4 2,200 0 19 2 1 2.0 294.0 2 2,100 1 13 2 1 2.5 176.3 2 2,000 0 17 3 0 2.0 294.3 7 2,400 1 8 4 1 2.0 224.0 3 1,900 0 6 1 1 2.0 125.0 2 1,900 1 18 4 0 1.5 236.8 4 2,600 0 17 5 1 2.0 164.1 4 2,300 1 19 4 0 2.0 217.8 3 2,500 1 12 3 0 2.0 192.2 2 2,400 1 16 2 0 2.5 125.9 2 2,400 1 28 1 0 1.5 220.9 2 2,300 0 12 1 1 2.0 294.5 6 2,700 1 15 3 1 2.0 244.6 2 2,300 1 9 2 1 2.5 199.0 3 2,500 0 18 1 0 1.5 240.0 4 2,600 1 13 4 1 2.0 264.2 4 2,300 1 14 3 1 2.0 188.1 2 1,900 1 8 4 1 1.5 243.7 6 2,700 1 7 4 1 2.0 221.5 4 2,300 1 18 3 1 2.0 175.0 2 2,500 1 11 3 0 2.0 253.2 3 2,300 1 16 2 1 2.0 155.4 4 2,400 0 16 3 0 2.0 186.7 5 2,500 0 21 4 0 2.5 179.0 3 2,400 0 10 4 1 2.0 188.3 6 2,100 0 15 4 1 2.0 227.1 4 2,900 1 8 4 1 2.0 173.6 4 2,100 0 14 2 1 2.5 188.3 5 2,300 1 20 5 0 1.5 310.8 8 2,900 1 9 4 1 2.0 293.7 6 2,400 1 11 5 1 3.0 179.0 3 2,400 1 8 4 1 2.0 188.3 6 2,100 0 14 2 1 2.5 227.1 4 2,900 1 20 5 0 1.5 173.6 4 2,100 1 9 4 1 2.0 188.3 5 2,300 1 11 5 1 3.0 A.1 Data Set 1-Goodyear, Arizona, Real Estate Sales Data Variables x, = Selling price in $000 = Num ber of bedrooms x3Size of the home in square feet x, = Pool (1-yes, 0 = no) Distance from the center of the city in miles = Township x, = Garage attached (1 = yes, 0 = no) = Number of bathrooms 105 homes soldExplanation / Answer
(a)
n = 105
= 220
s = 47.11
x-bar = 221.1
Hypotheses:
Ho: 220
Ha: > 220
Decision Rule:
= 0.01
Critical z- score = 2.326347874
Reject Ho if z > 2.326347874
Test Statistic:
SE = s/Ön = 47.11/105 = 4.597465244
z = (x-bar - )/SE = (221.1 - 220)/4.59746524366054 = 0.239262276
p- value = 0.405451108
Decision (in terms of the hypotheses):
Since 0.239262276 < 2.326347874 we fail to reject Ho
Conclusion (in terms of the problem):
No, we can’t conclude that the mean selling price is more than $220000.
(b)
n = 105
= 2100
s = 248.66
x-bar = 2224
Hypotheses:
Ho: 2100
Ha: > 2100
Decision Rule:
= 0.01
Critical z- score = 2.326347874
Reject Ho if z > 2.326347874
Test Statistic:
SE = s/Ön = 248.66/105 = 24.26673121
z = (x-bar - )/SE = (2224 - 2100)/24.2667312139382 = 5.109876518
p- value = 1.61185E-07
Decision (in terms of the hypotheses):
Since 5.109876518 < 2.326347874 we fail to reject Ho
Conclusion (in terms of the problem):
Yes, we can conclude that the mean size of homes is more than 2100 sq ft
(c)
Out of 105 homes, 71 have attached garage
n = 105
p = 0.6
p' = 71/105 = 0.6761905
Hypotheses:
Ho: p 0.6
Ha: p > 0.6
Decision Rule:
= 0.05
Critical z- score = 1.644853627
Reject Ho if z > 1.644853627
Test Statistic:
SE = Ö{(p (1 - p)/n} = (0.6 * (1 - 0.6)/105) = 0.047809144
z = (p' - p)/SE = (0.676190476190476 - 0.6)/0.0478091443733757 = 1.593638146
p- value = 0.0555086
Decision (in terms of the hypotheses):
Since 1.5936381 < 1.644853627 we fail to reject Ho
Conclusion (in terms of the problem):
No, we can’t conclude that more than 60% of the homes have attached garage
(d)
Out of 105 homes, 67 had a pool
n = 105
p = 0.4
p' = 67/105 = 0.638095
Hypotheses:
Ho: p ³ 0.4
Ha: p < 0.4
Decision Rule:
= 0.05
Critical z- score = -1.6449
Reject Ho if z < -1.6449
Test Statistic:
SE = Ö{p (1 - p)/n} = (0.4 * (1 - 0.4)/105) = 0.0478
z = (p' - p)/SE = (0.638095238095238 - 0.4)/0.0478091443733757 = 4.9801
p- value = 1.0000
Decision (in terms of the hypotheses):
Since 4.9801 > -1.6449 we fail to reject Ho
No, we can’t conclude that less than 40% of the homes had a pool
62. Refer to the Real Estate data, which report information on homes sold in Phoenix, Arizona, last year.
a. Let selling price be the dependent variable and size of the home the independent variable. Determine the regression equation. Estimate the selling price for a home with an area of 2,200 square feet. Determine the 95 percent confidence interval and the 95 percent prediction interval for the selling price of a home with 2,200 square feet.
b. Let selling price be the dependent variable and distance from the center of the city the independent variable. Determine the regression equation. Estimate the selling price of a home 20 miles from the center of the city. Determine the 95 percent confidence interval and the 95 percent prediction interval for homes 20 miles from the center of the city.
c. Can you conclude that the independent variables “distance from the center of the city” and “selling price” are negatively correlated and that the area of the home and the selling price are positively correlated? Use the .05 significance level. Report the p-value of the test.
(a) Regression Output:
The regression equation is Price = 64.7931 + 0.0703 * Size
Estimated selling price of a home with area of 2200 sq ft = $219429
95% confidence interval for the selling price of a home with area of 2200 sq ft is [$210883, $227976]
95% prediction interval for the selling price of a home with area of 2200 sq ft is [$131878, $307021]
(b) Regression Output:
The regression equation is Price = 270.167 - 3.354 * Distance
Estimated selling price of a home 20 miles from the city center = $203087
95% confidence interval for the selling price of a home 20 miles from the city center is [$190267, $215907]
95% prediction interval for the selling price of a home 20 miles from the city center is [$114118, $292057]
(c)
Test for negative correlation between Selling price and Distance:
p- value for this test (as seen from the regression output) is 0.0003. Since this is < 0.05, we can conclude that Selling price and Distance from the city center are negatively correlated.
Test for positive correlation between Selling price and Size:
p- value for this test (as seen from the regression output) is 0.0001. Since this is < 0.05, we can conclude that Selling price and Size are positively correlated.
(a)
n = 105
= 220
s = 47.11
x-bar = 221.1
Hypotheses:
Ho: 220
Ha: > 220
Decision Rule:
= 0.01
Critical z- score = 2.326347874
Reject Ho if z > 2.326347874
Test Statistic:
SE = s/Ön = 47.11/105 = 4.597465244
z = (x-bar - )/SE = (221.1 - 220)/4.59746524366054 = 0.239262276
p- value = 0.405451108
Decision (in terms of the hypotheses):
Since 0.239262276 < 2.326347874 we fail to reject Ho
Conclusion (in terms of the problem):
No, we can’t conclude that the mean selling price is more than $220000.
(b)
n = 105
= 2100
s = 248.66
x-bar = 2224
Hypotheses:
Ho: 2100
Ha: > 2100
Decision Rule:
= 0.01
Critical z- score = 2.326347874
Reject Ho if z > 2.326347874
Test Statistic:
SE = s/Ön = 248.66/105 = 24.26673121
z = (x-bar - )/SE = (2224 - 2100)/24.2667312139382 = 5.109876518
p- value = 1.61185E-07
Decision (in terms of the hypotheses):
Since 5.109876518 < 2.326347874 we fail to reject Ho
Conclusion (in terms of the problem):
Yes, we can conclude that the mean size of homes is more than 2100 sq ft
(c)
Out of 105 homes, 71 have attached garage
n = 105
p = 0.6
p' = 71/105 = 0.6761905
Hypotheses:
Ho: p 0.6
Ha: p > 0.6
Decision Rule:
= 0.05
Critical z- score = 1.644853627
Reject Ho if z > 1.644853627
Test Statistic:
SE = Ö{(p (1 - p)/n} = (0.6 * (1 - 0.6)/105) = 0.047809144
z = (p' - p)/SE = (0.676190476190476 - 0.6)/0.0478091443733757 = 1.593638146
p- value = 0.0555086
Decision (in terms of the hypotheses):
Since 1.5936381 < 1.644853627 we fail to reject Ho
Conclusion (in terms of the problem):
No, we can’t conclude that more than 60% of the homes have attached garage
(d)
Out of 105 homes, 67 had a pool
n = 105
p = 0.4
p' = 67/105 = 0.638095
Hypotheses:
Ho: p ³ 0.4
Ha: p < 0.4
Decision Rule:
= 0.05
Critical z- score = -1.6449
Reject Ho if z < -1.6449
Test Statistic:
SE = Ö{p (1 - p)/n} = (0.4 * (1 - 0.4)/105) = 0.0478
z = (p' - p)/SE = (0.638095238095238 - 0.4)/0.0478091443733757 = 4.9801
p- value = 1.0000
Decision (in terms of the hypotheses):
Since 4.9801 > -1.6449 we fail to reject Ho
No, we can’t conclude that less than 40% of the homes had a pool
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