Question

The CDC reports the probability a 25 year old adult will survive to age 35 is 0.986. you select twenty 25-year-old adults at random.

A) What is the probability exactly 19 adults survive to age 35? All 20 survive? At least 19 survive?

B) What is the probability at most 5 adults pass away before 35? at least 15 survive?

C) Find the expected value and explain what it means in context

D) Find the minimum number of 25 year old adults we need to select at random for the probability for the probabilty distribution to be approximatley normal.

E) Under the conditions of part (D) find the upper and lower fences

Answer 1

Solution

Let X = number of 25 year old adults who will survive to age 35, out of a random sample of twenty 25-year-old adults.

Then, X ~ B(20, 0.986), where 0.986 = probability a 25 year old adult will survive to age 35 [given]

Back-up Theory

If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where n = number of trials and p = probability of one success, then

probability mass function (pmf) of X is given by

p(x) = P(X = x) = (ⁿC_x)(p^x)(1 - p)^{n – x}, x = 0, 1, 2, ……. , n ……………………………………………………….………….............................................................…..(1)

[The above probability can also be directly obtained using Excel Function of Binomial Distribution: BINOMDIST(Number_s:Trials:Probability_s:Cumulative), what is within brackets is (x:n:p:True)] ….(1a)

Mean (average) of X = E(X) = µ = np........................………….…………………………………………..(2)

If X ~ B(n, p), np ≥ 5 and np(1 - p) ≥ 5, then Binomial probability can be approximated by Standard Normal probabilities ......................................................................................................................…..(3)

Now to work out the solution,

Part (a)

Probability exactly 19 adults survive to age 35 = P(X = 19) = 0.2142 [vide (1a)] ANSWER 1

Probability all 20 adults survive to age 35 = P(X = 20) = 0.75429 [vide (1a)] ANSWER 2

Probability at least 19 adults survive to age 35

= P(X = 19) + P(X = 20) = 0.96849 [sum of Answer 1 and Answer 2] ANSWER 3

Part (b)

Probability at most 5 adults pass out before age 35

= Probability at least 15 adults survive to age 35

= P(X ≥ 15)

= 1 [vide (1a)] ANSWER 4

Part (c)

Vide (3), expected value = 20 x 0.986 = 19.72 ANSWER 5

The above means that on an average 19.72 out of 20 are likely to be alive up to age 35. ANSWER 6

Part (d)

Vide (3), n x 0.986 ≥ 5 => n ≥ 5 and n x 0.986 x 0.014 ≥ 5 => n ≥ 362.

Thus, at least 362 person must be sampled. ANSWER 7

DONE