Question

Question 4: teens and Pocket money a) If 30% of teenagers take their pocket money from part time jobs. Create a binomial probability table with cumulative probabilities for a sample of 15 teenagers; b) Find the probability that exactly 10 have part time jobs. c) Find the probability that at least 8 have part time jobs. d) Find the probability that more than 5 do not have part time jobs. e) Graph the distribution. And describe its shape

Answer 1

Given n = 15, p = 0.30

a)

The pmf of Binomial distribution is PO p" (1 - p)*, x = 0,1,2,...,n The distribution of X is binomial distribution i.e. X – B(n, p) where n and p are paramters where n is number of trails and p is probability of success Given n =15 and p = 0.30 Let X be the number of part time jobs The probability distribution fucntion of X is 15 (0.30) (1 -0.30) S-*, x = 0,1,...,15

Substituting x values in P(X=x) then we get probabilities and also calculated cumulative probabilities F(x) as follows

x	P(X=x)	F(x) = P(X<=x)
0	0.004748	0.004748
1	0.030520	0.035268
2	0.091560	0.126828
3	0.170040	0.296868
4	0.218623	0.515491
5	0.206130	0.721621
6	0.147236	0.868857
7	0.081130	0.949987
8	0.034770	0.984757
9	0.011590	0.996347
10	0.002980	0.999328
11	0.000581	0.999908
12	0.000083	0.999991
13	0.000008	0.999999
14	0.000001	1.000000
15	0.000000	1.000000

b) The probability that exactly 10 have part time jobs is 0.002980

i.e. P(X=10 ) = 0.002980

c) P(X >= 8) = 1 - P(X < 8)

= 1 - P(X <= 7)

= 1 - F(7)

= 1 - 0.949987

= 0.050013

d) P(More than 5 do not have part time jobs)
= P(X < = 5) = F(5) = 0.721621

e) Graph:

0.25 0.20 0.15 (X=Xd 0.10 0.05 0.00 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

The shape of the histogram is positive skewed or right skewed distribution

since curve increases fastly and decreases slowly.