Question

Decide whether you can use the normal distribution to approximate the binomial distribution. If you can use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot explain why and use the binomial distribution to find the indicated probabilities A survey of adults found that 8% say their favorite sport is auto racing. You randomly select 600 adults and ask them to name their favorite sport. Complete parts (a) through (d). Determine whether a normal distribution can be used to approximate the binomial distribution. Choose the correct answer below. O A. No, because ng < 5. OB. No, because np < 5. O c. Yes, because both np 25 and ng 25 (a) Find the probability that the number of people who say auto racing is their favorite sport is at most 56 (Round to four decimal places as needed.) Sketch the graph of the normal distribution with the indicated probability shaded. ОА OB Ос. 48 The normal distribution cannot be used to approximate the binomial distribution. Click to select your answer(s).

Answer 1

p = 0.08, n = 600

Thus, np = 48 and nq = 552

Normal distribution can be used to approximate the binomial distribution.

Yes, because both np ≥ 5 and nq ≥ 5

Let X denote the number of adults who say their favourite sport is auto racing

Thus, X can be approximated to Normal distribution with

Mean,. $\mu$ = np = 48

and Standard deviation, $\sigma$ = $\sqrt {npq}$ = 6.645

(a) The required probability = P(X ≤ 56)

Using correction of continuity, P(X ≤ 56) ≈ P(X < 56.5)

= P{Z < (56.5 - 48)/6.645} = P(Z < 1.28) = 0.8997

Graph in option (C) denotes the correct graph

(b) The required probability = P(X > 60)

Using correction of continuity, P(X > 60) ≈ P(X > 60.5)

= P(Z > 1.88) = 0.03

Graph in option (B) denotes the correct graph

(c) The required probability = P(43 ≤ X ≤ 50)

Using correction of continuity, P(43 ≤ X ≤ 50) = P(42.5 < X < 50.5)

= P(-0.828 < Z < 0.376)

= 0.4426

Graph in option (B) denotes the correct graph

(d) The event in part (b) is unusual because it's Probability is less than 0.05