The distribution of income of families in the US in 1979 (in actual dollars uncorrected for in?ation) is shown in the table below:
|
Income level,x, (× $103 )
0-5
|
Percent of Population with income level, x 4
|
|
5-10
|
13
|
|
10-15
|
17
|
|
15-20
|
20
|
|
20-25
|
16
|
|
25-30
|
12
|
|
30-35
|
7
|
|
35-40
|
4
|
|
40-45
|
3
|
|
45-50
|
2
|
|
50-55
|
1
|
|
> 55
|
1
|
(i) Plot the data histogram and comment on the shape.
(ii) Using the center of the interval to represent each income group, determine the mean, median, mode; and the variance and skewness for this data set. Comment on how consistent the numerical values computed for these characteristics are with the shape of the histogram.
(iii) If the 1979 population is broadly classi?ed according to income into "Lower Class" for income range (in thousands of dollars) 0-15, "Middle Class" for income range, 15-50 and "Upper Class" for income range >50, what is the probability that two people selected at random and sequentially to participate in a survey from the Census Bureau (in preparation for the 1980 census) are (a) both from the "Lower Class," (b) both from the "Middle Class," (c) one from the "Middle class" and one from the "Upper class," and (d) both from the "Upper class"?
(iv) If, in 1979, engineers with at least 3 years of college education (excluding gradu- ate students) constitute approximately 1% of the population, (2.2 million out of 223 million) and span the income range from 20-55, determine the probability that an individual selected at random from the population is in the middle class given that he/she is an engineer. Determine the converse, that the person selected at random is an engineer given that he/she is in the middle class.