Question

Assume that the speed of automobiles on an expressway during rush hour is normally distributed with a mean of 61 mph and a standard deviation of 10 mph. If 300 cars are selected at random, how many will be traveling slower than 45 mph?

a) Calculate the z-score for a speed of 45 mph.
b) Use the z-score to find the proportion of cars traveling slower than 45 mph.
c) Calculate the number of cars out of 300 that will be traveling slower than 45 mph.
d) all of the above

Answer 1

To find the number of cars traveling slower than 45 mph, calculate the z-score for 45 mph, use the z-score to find the proportion of cars below 45 mph, and then calculate the number out of 300 cars that this proportion represents.

Step-by-step explanation:

The question involves applying the principles of normal distribution to determine the number of cars traveling below a certain speed.

1. Calculate the z-score for a speed of 45 mph: The z-score formula is Z = (X - μ) / σ, where X is the value in question, μ is the mean, and σ is the standard deviation. So, Z = (45 - 61) / 10 = -1.6.
2. Proportion of cars traveling slower than 45 mph: We look up the z-score of -1.6 on a standard normal distribution table or use a statistical software to find the proportion, which corresponds approximately to 0.0548.
3. Number of cars out of 300 traveling slower than 45 mph: Multiply the proportion by 300 to get the expected number. So, 300 * 0.0548 = 16.44, which we round to 16 cars.