Question

1. Suppose 20% of OSU students watch reality TV shows of some kind every week. Ten OSU students are selected at random and asked if they watch reality TV of some kind each week. You want the probability that at least 4 of them say yes.
   1. What does X represent in this problem?
   2. Show BRIEFLY that X has a binomial distribution.
   3. Find the probability that at least 4 of the students say yes.
   4. Could a person use the normal approximation to answer this question? Explain briefly.

Answer 1

a)

X represents number of students says Yes.

b)

Sample selected is random and probability of each outcome is fixed, 0.20

So X has binomial distribution.

c)

P(X) = ⁿC_x p^x ( 1 - p)^n-x

P(X >= 4) = 1 - P(X <= 3)

= 1 - [ P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) ]

= 1 - [ ¹⁰C₀ * 0.20⁰ * ( 1 - 0.20)¹⁰ + ¹⁰C₁ * 0.20¹ * ( 1 - 0.20)⁹ + ¹⁰C₂ * 0.20² * ( 1 - 0.20)⁸ + ¹⁰C₃ * 0.20³ * ( 1 - 0.20)³ ]

= 0.1209

d)

Normal approximation is appropriate if both np >= 5 and n( 1 - p) >= 5

np = 10 * 0.20 = 2 < 5

nq = 10 * ( 1 - 0.2) = 8 > 5

Since np < 5 , Normal approximation can not be used.