Question

1) If 20% of the residents in a U.S. city prefer a white telephone over any other color available, what is the probability that among the next 1000 telephones installed in that city (a) between 170 and 185 inclusive will be white? (b) at least 210 but not more than 225 will be white?

2) The serum cholesterol level X in 14-year-old boys has approximately a normal distribution with mean 170 and standard deviation 30. (a) Find the probability that the serum cholesterol level of a randomly chosen 14-year-old boy exceeds 230. (b) In a middle school there are 300 14-year-old boys. Find the probability that at least 8 boys have a serum cholesterol level that exceeds 230.

3) Suppose that the service life, in years, of a hearing aid battery is a random variable having a Weibull distribution with = 2 and β =1/2. ) (a) How long can such a battery be expected to last? (b) What is the probability that such a battery will be operating after 2 years?

Answer 1

Answer:

1.

Given,

n = 1000

p = 20% = 0.20

q = 1 - 0.20

= 0.80

Mean = np

= 1000*0.20

= 200

Standard deviation = sqrt(pq)

= sqrt(1000*0.20*0.80)

= 12.6491

a)

To give the probability P(170 <= x <= 185)

P(170 <= x <= 185) = P((169.5 - 200)/12.6491 <= (x-mu)/s <= (185.5 - 200)/12.6491)

= P(-2.41 <= z <= -1.15)

= P(z <= -1.15) - P(z <= -2.41)

= 0.1251 - 0.0080 [since from z table]

= 0.1171

b)

P(210 <= x <= 225)

= P((209.5 - 200)/12.6491 <= (x-mu)/s <= (225.5 - 200)/12.6491)

= P(0.75 <= z <= 2.02)

= P(z <= 2.02) - P(z <= 0.75)

= 0.9783 - 0.7734

= 0.2049

I hope it works for you.

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