Question

1) A coin is tossed 400 times. Use the normal curve approximation to find the probability of obtaining (a) between 185 and 210 heads inclusive; (b) exactly 205

heads; (c) fewer than 176 or more than 227 heads.

2) A process for manufacturing an electronic component yields items of which 1% are defective. A quality control plan is to select 100 items from the process, and if

none are defective, the process continues. Use the normal approximation to the binomial to find (a) the probability that the process continues given the sampling

plan described; (b) the probability that the process continues even if the process has gone bad (i.e., if the frequency of defective components has shifted to 5.0%

defective)

3) Researchers at George Washington University and the National Institutes of Health claim that approximately 75% of people believe “tranquilizers work very well to make a person more calm and relaxed.” Of the next 80 people interviewed, what is the probability that (a) at least 50 are of this opinion? (b) at most 56 are of this opinion?

Answer 1

Answer:

1.

Given,

n = 400

p = 0.5

q = 0.5

Mean = np

= 400*0.5

= 200

standard deviation = sqrt(npq)

= sqrt(400*0.5*0.5)

= 10

a)

To give the normal approximation probability between 185 & 210

x1 = 184.5 , x2 = 210.5

P(185 <= x <= 210) = P((184.5 - 200)/10 <= (x-mu)/s <= (210.5 - 200)/10)

= P(-1.55 <= z <= 1.05)

= P(z <= 1.05) - P(z <= -1.55)

= 0.8531 - 0.0606 [since from z table]

= 0.7926

b)

To give the normal approximation probability for exactly 205 heads

let us consider,

P(x = 205) = P((204.5 - 200)/10 <= (x-mu)/s <= (205.5 - 200)/10)

= P(0.45 <= z <= 0.55)

= P(z <= 0.55) - P(z <= 0.45)

= 0.7088 - 0.6736 [since from z table]

= 0.0352

c)

To give the normal approximation probability fewer than 176 or more than 227 heads

P(176 <= x <= 227) = P((175.5 - 200)/10 <= (x-mu)/s <= (227.5 - 200)/10)

= P(-2.45 <= z <= 2.75)

= P(z <= 2.75) - P(z <= -2.45)

= 0.9970 - 0.0071

= 0.9899

I hope it works for you.

Please post the remaining questions as separate post. Thank you.