Question

Assume that the speed of automobiles on an expressway during rush hour is normally distributed with a mean of 65 mph and a standard deviation of 5 mph. If 500 cars are selected at​ random, approximately how many will be traveling slower than 57 ​mph?

Hint. Calculate the​ z-score using the following formula. Then use the given tables to find the area that corresponds to the​ z-score. When determining the area to the left of a ​ z-score, use the appropriate table. When determining the area to the right of a​ z-score, subtract the percent of data to the left of the specified​ z-score from​ 100%. When determining the area between two​ z-scores, subtract the smaller area from the larger area. Then change the resulting area to a percent.

Answer 1

To find out how many cars will be traveling slower than 57 mph, the z-score of -1.6 is calculated using the given mean and standard deviation. About 5.48% of the cars are slower than 57 mph, hence approximately 27 cars out of 500 will be traveling at this speed or slower.

Step-by-step explanation:

The student is asking a question related to normal distribution in statistics, a common topic in both high school and college-level mathematics. To find how many cars will be traveling slower than 57 mph, we first need to calculate the z-score, which is the number of standard deviations an element is from the mean. The formula for the z-score is:

z = (X - μ) / σ

where X is the value we're interested in, μ is the mean, and σ is the standard deviation. For this example:

z = (57 - 65) / 5 = -1.6

Using the standard normal distribution tables, we can find the area to the left of the z-score of -1.6, which corresponds to approximately 0.0548 or 5.48%. This represents the proportion of cars traveling slower than 57 mph. To find the approximate number of cars:

Number of cars = total cars × proportion = 500 × 0.0548 ≈ 27

Therefore, approximately 27 cars out of 500 will be traveling slower than 57 mph during rush hour.