Answer:
Explanation:
A. The speed of cars on the freeway is normally distributed with a mean (μ) of 71 mph and a standard deviation (σ) of 9 mph. So, you can express this as:
X ~ N(71, 9^2)
B. To find the probability that a randomly selected car is traveling more than 70 mph, you can use the standard normal distribution table or a calculator to find the z-score for 70 mph and then find the probability associated with that z-score being less than or equal to it. The formula for the z-score is:
\[z = \frac{X - μ}{σ}\]
Where X is the value you want to find the probability for (in this case, 70 mph), μ is the mean (71 mph), and σ is the standard deviation (9 mph).
\[z = \frac{70 - 71}{9} = -\frac{1}{9} \approx -0.11\]
Now, find the probability using the z-score:
P(X > 70) = 1 - P(X ≤ 70)
Using a standard normal distribution table or calculator, you can find P(X ≤ -0.11) and subtract it from 1 to get the answer.
C. To find the probability that a randomly selected car is traveling between 73 and 79 mph, you can calculate the z-scores for both 73 and 79 mph and find the probabilities associated with those z-scores. Then, subtract the probability associated with the lower z-score from the probability associated with the higher z-score.
\[z_1 = \frac{73 - 71}{9} = \frac{2}{9} \approx 0.22\]
\[z_2 = \frac{79 - 71}{9} = \frac{8}{9} \approx 0.89\]
Now, find the probabilities:
P(73 ≤ X ≤ 79) = P(0.22 ≤ Z ≤ 0.89)
You can use a standard normal distribution table or calculator to find these probabilities.
D. To find the speed at which 63% of all cars travel on the freeway, you need to find the z-score that corresponds to the 63rd percentile of the standard normal distribution. You can then use the z-score formula to find the speed associated with that z-score.
First, find the z-score corresponding to the 63rd percentile, denoted as z_(0.63). You can use a standard normal distribution table or calculator to find this value.
Once you have the z_(0.63) value, you can use it to find the speed (X) using the z-score formula:
\[z_(0.63) = \frac{X - μ}{σ}\]
Solve for X:
\[X = z_(0.63) * σ + μ\]
Substitute the values of z_(0.63), σ, and μ to calculate X.