1. Assume the speed of vehicles along a stretch of I-10 has an approximately normal distribution with a mean of 71 mph and a standard deviation of 8 mph.

a. The current speed limit is 65 mph. What is the proportion of vehicles less than or equal to the speed limit?

b. What proportion of the vehicles would be going less than 50 mph?

c. A new speed limit will be initiated such that approximately 10% of vehicles will be over the speed limit. What is the new speed limit based on this criterion?

d. In what way do you think the actual distribution of speeds differs from a normal distribution?

2. A group of students at a school takes a history test. The distribution is normal with a mean of 25, and a standard deviation of 4. 

(a) Everyone who scores in the top 30% of the distribution gets a certificate. What is the lowest score someone can get and still earn a certificate? 

(b) The top 5% of the scores get to compete in a statewide history contest. What is the lowest score someone can get and still go onto compete with the rest of the state?

3. Use the normal distribution to approximate the binomial distribution and find the probability of getting 15 to 18 heads out of 25 flips. Compare this to what you get when you calculate the probability using the binomial distribution. Write your answers out to four decimal

The Standard Normal Distribution

Use the following information to answer the next two exercises: The patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.3 days and a standard deviation of 2.1 days.

4. What is the median recovery time?

a. 2.7

b. 5.3

c. 7.4

d. 2.1

5. Height and weight are two measurements used to track a child’s development. The World Health Organization measures child development by comparing the weights of children who are the same height and the same gender. In 2009, weights for all 80 cm girls in the reference population had a mean μ = 10.2 kg and standard deviation σ = 0.8 kg. Weights are normally distributed.  X ~ N(10.2, 0.8). Calculate the z-scores that correspond to the following weights and interpret them.

a. 11 kg

b. 7.9 kg

c. 12.2 kg

6. Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet.

a. If X = distance in feet for a fly ball, then X ~ _____(_____,_____)

b. If one fly ball is randomly chosen from this distribution, what is the probability that this ball traveled fewer than 220 feet? 

Sketch the graph. Scale the horizontal axis X. Shade the region corresponding to the probability. Find the probability.

c. Find the 80th percentile of the distribution of fly balls. Sketch the graph, and write the probability statement.

7. Facebook provides a variety of statistics on its Web site that detail the growth and popularity of the site. On average, 28 percent of 18 to 34 year olds check their Facebook profiles before getting out of bed in the morning. Suppose this percentage follows a normal distribution with a standard deviation of five percent.

a. Find the probability that the percent of 18 to 34-year-olds who check Facebook before getting out of bed in the morning is at least 30.

b. Find the 95th percentile, and express it in a sentence.

The Central Limit Theorem for Sample Means (Averages)

8. Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. We randomly sample 49 fly balls.

a. If X ¯ = average distance in feet for 49 fly balls, then X ¯ ~ _______(_______,_______)

b. What is the probability that the 49 balls traveled an average of less than 240 feet? Sketch the graph. Scale the horizontal axis for X ¯. Shade the region corresponding to the probability. Find the probability.

c. Find the 80th percentile of the distribution of the average of 49 fly balls.

9. Which of the following is NOT TRUE about the distribution for averages?

a. The mean, median, and mode are equal.

b. The area under the curve is one.

c. The curve never touches the x-axis.

d. The curve is skewed to the right.

10. A typical adult has an average IQ score of 105 with a standard deviation of 20. If 20 randomly selected adults are given an IQ test, what is the probability that the sample mean scores will be between 85 and 125 points?