Question

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

It is estimated that 3.6% of the general population will live past their 90th birthday. In a graduating class of 768 high school seniors, find the following probabilities. (Round your answers to four decimal places.)

(a) 15 or more will live beyond their 90th birthday

(b) 30 or more will live beyond their 90th birthday

(c) between 25 and 35 will live beyond their 90th birthday

(d) more than 40 will live beyond their 90th birthday

Answer 1

Let X be the number of persons live beyond 90th birthday

Using Normal approximation to Binomial

X follow Normal with mean = np = 768*0.036 = 27.648

and standard deviation =

Then

(a) To find P( X>15)

= P( z > -2.45)

= 0.9929 ( from standard normal table / z table)

Probability that 15 or more live beyond 90th birthday is 0.9929

(b)

To find P( X>30)

= P( z > 0.46 )

= 0.3228 ( from standard normal table / z table)

Probability that 30 or more live beyond 90th birthday is 0.3228

(c) To find (25 < X <35)

= P(-0.51 <z <1.42  )

= P( -0.51 < z <0) + P( 0 < z < 1.42 )

= 0.1950+0.4232 ( from standard normal table / z table)

= 0.6182

Probability that between 25 and 35  live beyond 90th birthday is 0.6182

(d)

To find P( X>40)

= P( z > 2.39)

= 0.0084 ( from standard normal table / z table)

Probability that more than 40 live beyond 90th birthday is 0.0084