Question

The Bahamas is a tropical paradise made up of 700 islands sprinkled over 100,000 square miles of the Atlantic Ocean. According to the figures released by the government of the Bahamas, the mean household income in the Bahamas is $37,803 and the median income is $33,550. A demographer decides to use the lognormal random variable to model this nonsymmetric income distribution. Let Y represent household income, where for a normally distributed X, Y = e^X. In addition, suppose the standard deviation of household income is $13,000. Use this information to answer the following questions. [You may find it useful to reference the z table.]

a. Compute the mean and the standard deviation of X. (Round your intermediate calculations to at least 4 decimal places and final answers to 4 decimal places.)

b. What proportion of the people in the Bahamas have household income above the mean? (Round your intermediate calculations to at least 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)

c. What proportion of the people in the Bahamas have household income below $22,000? (Round your intermediate calculations to at least 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)

d. Compute the 70th percentile of the income distribution in the Bahamas. (Round your intermediate calculations to at least 4 decimal places, “z” value to 3 decimal places, and final answer to the nearest whole number.)

Answer 1

a)

mean(ϴ)=

ln(μ/(√((σ2/μ2)+1)))=

10.4843

std dev(ω)=

√(ln((σ2/μ2)+1))=

0.3343

b)

proportion of the people in the Bahamas have household income above the mean =P(X>ln(37803)

=P(Z>(ln(37803)-10.483)/0.3343)

=P(Z>0.17)=0.4325

c)

proportion of the people in the Bahamas have household income below $22,000 =P(Z<(ln(22000)-10.483)/0.3343)

=P(Z<-1.45)=0.0735

d)

70th percentile of the income distribution in the Bahamas =e^{10.4843+0.52*0.3343} =42594.19 ~ 42594