Question

5. This problem is about the population of New York City. Given below is a joint-probability distribution of NYC based on results from the 2010 Census. Age Under 10 years old 10-19 years old 20-29 years old 30-39 years old 40-49 years old 50-59 years old 60-69 years old 70-79 years old 80 and over Male 0.063 0.069 0.077 0.076 0.072 0.057 0.035 0.018 0.010 Female 0.060 0.067 0.079 0.080 0.077 0.066 0.046 0.028 Y 0.020 Write the answers to each of the following (10) (a) Find the probability that an RSP is under 40 years old. (b) Find the probability that an RSP is at least 50 years old. (c) Find the probability that an RSP is at least 50 years old, given the person is male. (d) Estimate the median age of the population of NYC.

Answer 1

The given data is inputted in EXCEL here as:

Age	Male	Female
Under 10	0.063	0.06
10-19	0.069	0.067
20-29	0.077	0.079
30-39	0.076	0.08
40-49	0.072	0.077
50-59	0.057	0.066
60-69	0.035	0.046
70-79	0.018	0.028
80 and over	0.01	0.02

a) The probability that an RSP is under 40 years old is computed here as:

= Sum of the values in both the columns in the first 4 rows

= 0.063 + 0.06 + 0.069 + .... + 0.08

= 0.571

Therefore 0.571 is the required probability here.

b) The probability that an RSP is at least 50 years old is computed here as:

= 0.057 + 0.066 + ... + 0.02

= 0.28

Therefore 0.28 is the required probability here.

c) Given the person is male, probability that the RSP is at least 50 years old is computed here as:

= P( male and at least 50 years old ) / P( at least 50 years old )

= 0.12 / 0.28

= 0.4286

Therefore 0.4286 is the required probability here.

d) We first compute the probabilities for each age range along with the cumulative probabilities here as:

Age(x)	P(X = x)	P(X <= x)
Under 10	0.123	0.123
10-19	0.136	0.259
20-29	0.156	0.415
30-39	0.156	0.571
40-49	0.149	0.72
50-59	0.123	0.843
60-69	0.081	0.924
70-79	0.046	0.97
80 and over	0.03	1

As we have here:
P(X <= 29) = 0.415
P(X <= 39) = 0.571

Therefore the median age here is computed as:
= 29 + 10*(0.5 - 0.415)/(0.571 - 0.415) = 34.45

Therefore 34.45 is the required median age here.