Question
Statistics published by the National Highway Traffic Safety Administration and the National Safety Council show that on an average weekend night, 1 in 10 drivers is drunk. If 400 drivers are randomly checked the following Saturday night, what is the probability that the number of drunk drivers will be 400? a) less than 32? b) greater than 49? e) at least 35 but less than 47?
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Answer to a math question Statistics published by the National Highway Traffic Safety Administration and the National Safety Council show that on an average weekend night, 1 in 10 drivers is drunk. If 400 drivers are randomly checked the following Saturday night, what is the probability that the number of drunk drivers will be 400? a) less than 32? b) greater than 49? e) at least 35 but less than 47?
Timmothy
4.888 Answers
**
We solve each part using the binomial distribution formula and approximations where necessary.
a) less than 32?
P(X < 32) = \sum_{k=0}^{31} \binom{400}{k} (0.1)^k (0.9)^{400-k}
Using normal approximation:
\mu = np = 400 \times 0.1 = 40
\sigma = \sqrt{np(1-p)} = \sqrt{400 \times 0.1 \times 0.9} = \sqrt{36} = 6
Normal approximation to the binomial:
P(X < 32) \approx P\left(\frac{X - 40}{6} < \frac{32 - 40}{6}\right) = P\left(Z < -\frac{8}{6}\right) = P\left(Z < -1.33\right) \approx 0.0918
So:
P(X < 32) \approx 0.0918
b) greater than 49?
P(X > 49) = 1 - P(X \leq 49)
Using normal approximation:
P(X \leq 49) \approx P\left(\frac{X - 40}{6} \leq \frac{49 - 40}{6}\right) = P\left(Z \leq 1.5\right) \approx 0.9332
So:
P(X > 49) \approx 1 - 0.9332 = 0.0668
e) at least 35 but less than 47?
P(35 \leq X < 47) = P(X < 47) - P(X < 35)
Using normal approximation:
P(X < 47) \approx P\left(\frac{X - 40}{6} < \frac{47 - 40}{6}\right) = P\left(Z < 1.17\right) \approx 0.8790
P(X < 35) \approx P\left(\frac{X - 40}{6} < \frac{35 - 40}{6}\right) = P\left(Z < -0.83\right) \approx 0.2033
So:
P(35 \leq X < 47) = P(X < 47) - P(X < 35) \approx 0.8790 - 0.2033 = 0.6757
Therefore:
a) P(X < 32) \approx 0.0918
b) P(X > 49) \approx 0.0668
e) P(35 \leq X < 47) \approx 0.6757
We solve each part using the binomial distribution formula and approximations where necessary.
a) less than 32?
Using normal approximation:
Normal approximation to the binomial:
So:
b) greater than 49?
Using normal approximation:
So:
e) at least 35 but less than 47?
Using normal approximation:
So:
Therefore:
a)
b)
e)
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