Question

3. Let's think about the house price. According to the Case-Shiller Home Price Indices in August 2009, Chicago and San Francisco have following sample mean and population standard deviations (the sample mean was calculated by daily base, so the sample size was 30):

	CHICAGO	San Francisco
Sample Mean	130.55	132.47
Population Standard Deviation	9	12

               

1. Using hypothesis test, prove if these house price indices are same. (Setup a hypothesis, show your works to perform the test, and state your verdict)
2. Some people argue that San Francisco has higher house price than that of Chicago. Prove/disprove the argument using a hypothesis test.
3. Let’s assume the population standard deviations are unknown, and the sample standard deviation of for Chicago is 9.2 and that of San Francisco is 11.5. Some people argue that San Francisco has higher variability (higher variance) in house prices than that of Chicago. Setup a hypothesis, perform the test and prove/disprove the argument.

Answer 1

Ho :   µ1 - µ2 =   0          
Ha :   µ1-µ2 ╪   0          
                  
Level of Significance ,    α =    0.05          
                  
sample #1   ------->              
mean of sample 1,    x̅1=   130.55          
population std dev of sample 1,   σ1 =    9          
size of sample 1,    n1=   30          
                  
sample #2   --------->              
mean of sample 2,    x̅2=   132.47          
population std dev of sample 2,   σ2 =    12          
size of sample 2,    n2=   30          
                  
difference in sample means = x̅1 - x̅2 =    130.55   -   132.47   =   -1.92
                  
std error , SE =    √(σ1²/n1+σ2²/n2) =    2.7386          
                  
Z-statistic = ((x̅1 - x̅2)-µd)/SE =    -1.92   /   2.7386   =   -0.7011
                  
  
p-value =        0.483250050   [excel formula =2*NORMSDIST(z)]      
Desison:   p-value>α , Do not reject null hypothesis              
Conclusion:   There is not enough evidence to conclude that   house price indices are not same

b)

   Sample 1:   San francisco
Sample Variance,    s₁² =    132.25
Sample size,    n₁ =    30
   Sample 2:   Chicago
Sample Standard deviation,    s₂² =    84.64
Sample size,    n₂ =    30

Null and alternative hypothesis:  
Hₒ : σ₁² = σ₂²  
H₁ : σ₁² > σ₂²  
Test statistic:  
F = s₁² / s₂² = 132.25 / 84.64 =    1.5625

Degree of freedom:  
df₁ = n₁-1 =    29
df₂ = n₂-1 =    29
  
P-value :  
P-value = F.DIST.RT(1.5625, 29, 29) =    0.1177

p value >α

there is not enough evidence that  San Francisco has higher variability (higher variance) in house prices than that of Chicago