Homework 7 Name: ID# Section 1 Find the probabilities for each of the following using the standard normal distribution. 1. P(0 < z < 1.69) 2. P(-1.57 < z < 0) 3. P(z > 1.16) 4. P(z < -1.77) 5. P(-2.46 < z < 1.74) 6. P(1.46 < z < 2.97) 7. P(z > -1.39) 8. P(z < 1.77) 9. P(-1.46 < z < -0.97) 1
2 Find the z value that corresponds to the given area. 1. 2. 3 Find the z value to the right of the mean so that 53.98% of the area under the distribution curve lies to the left of it. 4 Find the z value to the left of the mean so that 98.87% of the area under the distribution curve lies to the right of it. 2
5 Find two z values so that 44% of the middle area is bounded by them. 6 Find two z values, one positive and one negative, so that the areas in the two tails total the following values. a. 5% b. 10% c. 1% 7 A survey found that people keep their microwave ovens an average of 3.2 years. The standard deviation is 0.56 year. If a person decides to buy a new microwave oven, find the probability that he or she has owned the old oven for the following amount of time. Assume the variable is normally distributed. a. Between 2 and 3 years. b. less than 1.5 years 3
c. more than 3.2 years d. What percent of microwave ovens would be replaced if a warranty of 18 months were given? 8 A local medical research association proposes to sponsor a footrace. The average time it takes to run the course is 45.8 minutes with a standard deviation of 3.6 minutes. Assume the variable is normally distributed. If the association decides to include only the top 25% of the racers, what should be the cutoff time in the tryout run? Would a person who runs the course in 40 minutes qualify? 9 To help students improve their reading, a school district decides to implement a reading program. It is to be administered to the bottom 5% of the students in the district, based on the scores on a reading achievement exam. If the average score for the students in the district is 122.6, find the cutoff score that will make a student eligible for the program. The standard deviation is 18. Assume the variable is normally distributed. 10 The average age of Amtrak passenger train cars is 19.4 years. If the distribution of ages is normal and 20% of the cars are older than 22.8 years, find the standard deviation. 4
11 A mandatory competency test for high school sophomores has a normal distribution with a mean of 400 and a standard deviation of 100. a. The top 3% of students receive $500. What is the minimum score you would need to receive this award? b. The bottom 1.5% of students must go to summer school. What is the minimum score you would need to stay out of this group? 12 An instructor gives a 100-point examination in which the grades are normally distributed. The mean is 60 and the standard deviation is 10. If there are 5% A s and 5% F s, 15% B s and 15% D s, and 60% C s, find the scores that divide the distribution into those categories. 13 Assume that the mean systolic blood pressure of normal adults is 120 millimeters of mercury (mm Hg) and the standard deviation is 5.6. Assume the variable is normally distributed. a. If an individual is selected, find the probability that the individual s pressure will be between 120 and 121.8 mm Hg. b. If a sample of 30 adults is randomly selected, find the probability that the sample mean will be between 120 and 121.8 mm Hg. c. Why is the answer to part a so much smaller than the answer to part b? 5
14 Determine whether each statement is true or false. If the statement is false, explain why. 1. The total area under a normal distribution is infinite. 2. The standard normal distribution is a continuous distribution. 3. All variables that are approximately normally distributed can be transformed to standard normal variables. 4. The z value corresponding to a number below the mean is always negative. 5. The area under the standard normal distribution to the left of z = 0 is negative. 6. The central limit theorem applies to means of samples selected from different populations. 15 Select the best answer. 1. The mean of the standard normal distribution is a. 0 b. 100 c. 1 d. Variable 2. Approximately what percentage of normally distributed data values will fall within 1 standard deviation above or below the mean? a. 68% b. 95% c. 99.7% d. Variable 3. Which is not a property of the standard normal distribution? a. It s symmetric about the mean. b. It s uniform. c. It s bell-shaped. d. It s unimodal. 4. When a distribution is positively skewed, the relationship of the mean, median, and mode from left to right will be a. Mean, median, mode b. Mode, median, mean c. Median, mode, mean d. Mean, mode, median 5. The standard deviation of all possible sample means equals a. The population standard deviation. b. The population standard deviation divided by the population mean. c. The population standard deviation divided by the square root of the sample size. d. The square root of the population standard deviation. 16 Complete the following statements with the best answer. 1. When one is using the standard normal distribution, P(z < 0) =. 2. The difference between a sample mean and a population mean is due to. 3. The mean of the sample means equals. 4. The standard deviation of all possible sample means is called. 6