Question

suppose that the speed at which cars go on the freeway is normally distributed with mean 68 mph and standard deviation 5 miles per hour. a. if one car is randomly chosen, find the probability that is traveling more than 70 mph. select b. if one of the cars is randomly chosen, find the probability that it is traveling between 65 and 75 mph. select c. find the z - score of a car going 77 mph. select

Explanation:

Step1: Define given values

Mean $\mu = 68$, standard deviation $\sigma = 5$

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Part A

Step1: Calculate z-score for 70 mph

$z = \frac{x - \mu}{\sigma} = \frac{70 - 68}{5} = 0.4$

Step2: Find $P(Z > 0.4)$

$P(Z > 0.4) = 1 - P(Z \leq 0.4)$
Using z-table, $P(Z \leq 0.4) = 0.6554$, so $1 - 0.6554 = 0.3446$

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Part B

Step1: Calculate z-scores for 65 and 75

$z_1 = \frac{65 - 68}{5} = -0.6$, $z_2 = \frac{75 - 68}{5} = 1.4$

Step2: Find $P(-0.6 < Z < 1.4)$

$P(Z < 1.4) - P(Z < -0.6)$
Using z-table, $P(Z < 1.4) = 0.9192$, $P(Z < -0.6) = 0.2743$
$0.9192 - 0.2743 = 0.6449$

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Part C

Step1: Calculate z-score for 77 mph

$z = \frac{77 - 68}{5} = \frac{9}{5} = 1.8$

Answer:

A. 0.3446
B. 0.6449
C. 1.8