Question

Obtaining a​ driver's license at the earliest eligible age has been a teenage ritual that appears to be fading. A study performed by a​ university's transportation research institute recently found that

3131​%

of U.S. teenagers between the ages of 17 and 19 did not possess a​ driver's license. A random sample of

130130

​17-to-19-year-old teenagers was selected.

A. What is the probability that 45 or fewer teenagers of this sample had not obtained a driver’s license?

B. What is the probability that 50 or more teenagers of this sample had not obtained a driver’s license?

C. What is the probability that between 35 and 40 teenagers had not obtained a driver’s license?

D. Suppose that 28 teenagers from this sample had not obtained a driver’s license? Does this result support the findings reported by the University of Michigan Transportation Research Institute?

Answer 1

Answer:

a)

Given,

Sample n = 140

p = 0.30

q = 1 - p

= 0.30

= 0.70

standard deviation = sqrt(p(1-p)/n)

substitute values

= sqrt(0.3(1-0.3)/140)

= 0.0387

sample proportion p^ = x/n

substitute values

= 45/140

= 0.321

P(p^ <= 0.321) = P(z <= (0.321 - 0.30)/0.0387)

= P(z <= 0.54)

= 0.7054014 [since from z table]

= 0.7054

b)

p^ = x/n = 50/140 = 0.357

P(X > 0.357) = P(z > (0.357 - 0.30)/0.0387)

= P(z > 1.47)

= 0.0707809 [since from z table]

= 0.0708

c)

p1^ = x/n = 35/140 = 0.25

p2^ = x/n = 40/140 = 0.286

P(0.25 < p^ < 0.286) = P((0.25 - 0.30)/0.0387 < z < (0.286 - 0.30)/0.0387)

= P(-1.29 < z < -0.36)

= P(z < - 0.36) - P(z < -1.29)

= 0.3594236 - 0.0985253 [since from z table]

= 0.2609

d)

p^ = 28/140 = 0.2

P(p^ <= 0.2) = P(z < (0.2 - 0.30)/0.0387)

= P(z < -2.58)

= 0.00494 [since from z table]

= 0.0049