Question

A highway study of 10,000 vehicles that passed by a checkpoint found that their speeds were normally distributed, with a mean of 63 mph and a standard deviation of 7 mph. (a) how many of the vehicles had a speed of more than 70 mph

Answer 1

Using z-scores and the standard normal distribution, approximately 1,587 out of 10,000 vehicles had speeds exceeding 70 mph, calculated by the area under the normal curve for z>1.

Step-by-step explanation:

To determine how many vehicles had a speed of more than 70 mph, we need to make use of the given normal distribution. The mean speed is 63 mph with a standard deviation of 7 mph.

First, we find the z-score for a speed of 70 mph:

Z = (X - μ) / σ

Z = (70 - 63) / 7

Z = 1

Next, we consult the standard normal distribution table or use a calculator to find the probability corresponding to a z-score of 1. This gives us the area to the left of z=1. To find the area to the right (which represents speeds greater than 70 mph), we subtract this value from 1.

Assuming the area to the left of z=1 is approximately 0.8413, we calculate:

Area to the right of z=1 = 1 - 0.8413 = 0.1587

Now, we calculate the number of vehicles out of 10,000 with speeds greater than 70 mph:

Number of vehicles = Total vehicles * Area to the right of z=1

Number of vehicles = 10,000 * 0.1587

Number of vehicles = 1,587

Therefore, approximately 1,587 vehicles had a speed of more than 70 mph.

Answer 2

The question requires using the standard normal distribution to find how many vehicles were traveling at speeds higher than 70 mph from a given set of 10,000 vehicles with a mean speed of 63 mph and a standard deviation of 7 mph.

Step-by-step explanation:

The question involves applying the concepts of normal distribution in statistics to determine the number of vehicles that were traveling above a certain speed. Given that the mean speed is 63 mph and the standard deviation is 7 mph, we can find the z-score for the speed of 70 mph and then calculate the proportion of vehicles exceeding this speed using the standard normal distribution table. Finally, we'll multiply the proportion by the total number of vehicles to find the number of vehicles with speeds over 70 mph.