According to the census, the median household income in Atlanta (1.5 Million hou

Question

According to the census, the median household income in Atlanta (1.5 Million households) was $52,000 in 1999. In June 2003, a market research organization takes a simple random sample of 750 households in Atlanta; 56% of the sample households had incomes over $52,000. Did median household income in Atlanta increase over the period 1999 to 2003?

a.) Formulate null and alternative hypotheses in terms of a box model.
b.) Calculate the appropriate test statistic and P.
c.) Did median family income go up?

Explanation / Answer

Since we are testing for proportion, a z-test is OK. Also, we have a large sample.
a)
H0: Median income in 2003 is $52,000 ( p=0.5)
HA: Median income is over $52,000 (p > 0.5)
p = proportion of people with income over $52,000 in 2003

b)
Sample proportion phat = 0.56
Variance of proportion = p*(1-p)/n
= 0.5(0.5)/750 =0.0003333
S.D. of p is sqrt[0.000333] = 0.0183
Z = ( 0.56 - 0.5 ) / 0.0183 = 3.2863

c)
P( z > 3.29) = 0.00001
The critical z assuming a 5 % level of significance is 1.645.
3.29 > 1.645; hence, reject H0 and accept HA.
The proportion of people with median income $52,000 is greater than 50 %.

Note:
We need to know in 1999, what percentage of people had income over $52,000 to precisely do a statistical test. (or at least a sample estimate taken in 1999).

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