Question

It is believed that approximately 75% of American youth now have insurance due to the health care law. Suppose this is true, and let X equal the number of American youth in a random sample of n = 15 with private health insurance. (a) How is X distributed? (b) Find the probability that X is at least 10. (c) Find the probability that X is at most 10. (d) Find the probability that X is equal to 10. (e) Give the mean, variance, and standard deviation of X

Answer 1

Answer:

Step-by-step explanation:

Binomial distribution can be used in this case

P(x=r) nCr×q^n-r×q^r

n =15

p = 75/100= 0.75

q= 1-p = 0.25

a) The distribution of x is discrete

b) the probability that x is at least 10

= probability of x greater than or equal to 10 = P(x= 10) +P(x = 11) + P(x=13) +P(x=14) +P(x =15)

P(x= 10) =15C10×0.25^5×0.75^10=0.1651

P(x = 11) =15C11×0.25^4×0.75^11=0.2252

(x=12) =15C1×0.25^3×0.75^12=0.2251

(x=13) =15C13×0.25^2×0.75^13=0.1559

(x=14)=15C14×0.25^1×0.75^14=0.0668 (x=15)=15C15×0.25^0×0.75^15=0.0134

probability of x greater than or equal to 10 = P(x= 10) +P(x = 11) + P(x=13) +P(x=14) +P(x =15)= 0.1651+ 0.2252+0.2251+0.1559+0.0668+0.0134

= 0.8381

c)

The probability that X is at most 10 =

1-probability of x is greater than or equal to 11

= 1- (0.2252+0.2251+0.1559+0.0668+0.0134 =1-0.673

=0.327

d)P(x=10)=0.1651

Mean = np =15*0.75=15

Standard deviation = npq=15×0.25=2.8125

Variance =2.8125^2=7.91