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.

a.

**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/(√((σ^{2}/μ^{2})+1)))= |
10.4843 |

std dev(ω)= |
√(ln((σ^{2}/μ^{2})+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

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