Question

The mean speed of a sample of vehicles along a stretch of highway is 67 miles per​ hour, with a standard deviation of 3 miles per hour. Estimate the percent of vehicles whose speeds are between 58 miles per hour and 76 miles per hour.​ (Assume the data set has a​ bell-shaped distribution.)

Answer 1

Answer: 99.9%

Explanation:In a normal distribution (bell-shaped distribution), the percent that is between the mean and the standard deviations are:

between the mean and mean + standard deviation the percentage is = 34.1%

between the mean + standard deviation and mean + 2 times the standard deviation is = 13.6%

between the mean + 2 times the standard deviation and the mean + 3 times the standard deviation is: 2.14%

And is the same if we subtract the standard deviation.

So in the range from 58 to 67, we can find 3 standard deviations, and in the range from 67 to 76, we also can find 3 standard deviations:

58 + 3 + 3 + 3 = 67

67 + 3 + 3 + 3 = 76

So the total probability is equal to the addition of all those ranges:

2.14% + 13.6% + 34.1% + 34.1% + 13.6% + 2.14% = 99.9%

So 99.9% of the cars have velocities in the range between 58 miles per hour and 76 miles per hour

Answer 2

99.85%

Explanation:

Most of today's student calculators have probability distribution functions built in. Here we are to find the area under the standard normal curve between 58 mph and 76 mph, if the mean speed is 67 mph and the std. dev. is 3 mph.

Here's what I'd type into my calculator:

normalcdf(58, 76, 67, 3)

The result obtained in this manner was 0.9985.

This states that 99.85% of the vehicles clocked were traveling at speeds between 58 mph and 76 mph.