According to theU.S. Census Bureau, the mean household income in the United Stat

Question

According to theU.S. Census Bureau, the mean household income in the United Statesin 2000 was $57,045 and the median household income was $42,148(U.S. Census Bureau, “Money Income in the United States:2000,” www.census.gov, September 2001). The variabilityof household income is quite large, with the 90th percentile (top10% of all households) approximately equal to $111,600, and anoverall standard deviation of approximately $25,000. Supposerandom samples of 225 households were selected.

a)       What proportion of thesample means would be below $55,000?

b)       What proportion of thesample means would be above $60,000?

c)       What proportion of thesample means would be above $111,600?

d)       Why is the probability youcalculated in (c) so much lower than 0.10, even though 10% ofindividual households have incomes above $111,600?

e)       If random samples of size 20were selected, can you use the methods discussed in this chapter tocalculate the probabilities requested in (a)--(c)? Explain.

Explanation / Answer

According to theU.S. Census Bureau, the mean household income in the United Statesin 2000 was $57,045 and the median household income was $42,148(U.S. Census Bureau, “Money Income in the United States:2000,” www.census.gov, September 2001). The variabilityof household income is quite large, with the 90th percentile (top10% of all households) approximately equal to $111,600, and anoverall standard deviation of approximately $25,000. Supposerandom samples of 225 households were selected.

a)       What proportion of thesample means would be below $55,000?

z = (55000-57045)/(25000/(225)) = -1.23 ==> P(z<-1.23)= 0.1093

b)       What proportion of thesample means would be above $60,000?

z = (60000-57045)/(25000/(225)) = 1.77 ==> P(z>1.77) =1-0.9616 = 0.0384

c)       What proportion of thesample means would be above $111,600?

z = (111600-57045)/(25000/(225)) = 32.73 ==>P(z>32.73) = 0

d)       Why is the probability youcalculated in (c) so much lower than 0.10, even though 10% ofindividual households have incomes above $111,600?

Because sample means vary less than individual incomes.

e)       If random samples of size 20were selected, can you use the methods discussed in this chapter tocalculate the probabilities requested in (a)--(c)? Explain.

No, the methods only work for large sample sizes.

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