Even before the Ownership Society programs of Presidents Cli

Even before the \\Ownership Society\" programs of Presidents Clinton and Bush, the federal
government was heavily involved in the housing market. In a study just prior to the foreclosure
crisis starting in 2008, the Urban Institute started about 80% of the estimated $200 billion
of federal housing subsidies consisted of tax breaks. Samples indicated that federal housing
benefits average $8,268 for those with incomes $200,000 and $500,000 and only $365 for those
with incomes of $40,000 to $50,000. The respective standard deviations were $2,100 and $150.
They were obtained from sample sizes of 150.

(a) To determine the appropriate test concerning the average federal housing benefits of the
two income groups, it is necessary to determine if the population variances are equal.
Conduct a test of hypothesis to this effect using = 0:02. Make sure to state the null and
the alternative hypotheses.

(b) Using the result from the statistical test obtained in part (a), test whether the benefits
to the high income group are at least $7,500 more than the bene ts to the low income
group using = 0:05. Make sure to state the null and the alternative hypotheses. Assume
bene ts are normally distributed for both income groups.

Solution

Here we have to test the hypothesis that

H0 : µ1 = µ2 Vs H1 : µ1 µ2

X1bar = 8268

X2bar = 365

s1 = 2100

s2 = 150

n1 = 150

n2 = 150

alpha = 0.02

Also given that population variances are equal.

Therefore we used here pooled standard deviation formula.

S = sqrt [ (n1-1)s1^2 + (n2-1)s2^2 / n1+n2-2 ]

S = sqrt [ (150-1) * 2100² + (150-1) * 150^2 / 150+150-2 ]

S = 1488.7075

The test statistic for testing is,

t = (X1bar - X2bar) / S * sqrt[ 1/n1 + 1/n2 ]

= (8268 - 365) / 1488.7075 sqrt [ 1/150 + 1/150 ]

= 45.9741

P-value we can calculate by using EXCEL syntax is,

=tdist(x, d.f.,tails)

where x is the test statistic value

d.f. = n1 +n2-2 = 150 + 150 -2 = 298

tails = 2

P-value = 0.0000

P-value < alpha

reject H0 at 2% level of significance.