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Practice test 3A, Normal Probability Distribution

The document covers various problems related to normal probability distributions, including calculations of probabilities, z-values, and properties of distributions. It also explores the application of normal distributions to different scenarios such as IQ scores, head breadths, and sample means. Additionally, it discusses the implications of the Central Limit Theorem and addresses real-world examples like determining the safety of boat weight limits.

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1 Statistics Sample Test Solution Chapter 6: Normal Probability Distributions 1. What percent of normally distributed data value lie within 2 standard deviations to either side of the mean? 2. For a continuous random variable, the probability of a single value of x is always _ 3. The parameters of normal distribution are ____________ 4. What are the values of mean and the standard deviation for the standard normal distribution? 5. Find the value of z for a standard normal distribution, such that the area in the left tail is 0.05 6. Find the value of z for a standard normal distribution, such that the area in the right tail is 0.1. 7. Find the following probability: P(z > -0.82) 8. The z-value for μ of normal distribution curve is always ________ 9. Usually the normal distribution is used as an approximation to binomial distribution when _________ 10. Three out of eight drinkers acquire the habit by age 20. If 300 drinkers are randomly selected, using normal approximation to the binomial distribution, find the probability that: a. Exactly 100 acquire the habit by age 20 b. At least 125 acquired the habit by age 20 c. At most 165 acquired the habit by age 20 11. IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. a. What percentage of people has an IQ score above 120? b. What is the maximum IQ score for the bottom 30%? 12. Men have head breaths that are normally distributed with a mean of 6.0 in and a standard deviation of 1.0 in. a. If a man is randomly selected, find the probability that his head breadth is less than 6.2 in. b. If 100 men are randomly selected, find the probability that their head breadth mean is less than 6.2 in.
2 13. Find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1. 14. Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of 1. Find the probability that a given score is between −2.13 and 3.77 and draw a sketch of the region. 15. Assume that a randomly selected subject is given a bone density test. Bone density test scores are normally distributed with a mean of 0 and a standard deviation of 1. Draw a graph and find P19, the 19th percentile. This is the bone density score separating the bottom 19% from the top 81 %. 16. Find the area of the shaded region. The graph to the right depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15.
3 17. The assets (in billions of dollars) of the four wealthiest people in a particular country are 33, 29, 16, 11. Assume that samples of size n =2 are randomly selected with replacement from this population of four values. a. After identifying the 16 different possible samples and finding the mean of each sample, construct a table representing the sampling distribution of the sample mean. In thetable, values of the sample mean that are the same have been combined. b. Compare the mean of the population to the mean of the sampling distribution of the sample mean. c. Do the sample means target the value of the populationmean? Ingeneral, do sample means make good estimates of population means? Why or whynot? 18. The population of current statistics students has ages with mean μ and standard deviation σ. Samples of statistics students are randomly selected so that there are exactly 44 students in each sample. For each sample, the mean age is computed. What does the central limit theorem tell us about the distribution of those mean ages? 19. A boat capsized and sank in a lake. Based on an assumption of a mean weight of 137 lb., the boat was rated to carry 50 passengers(so the load limit was 6,850 lb.). After the boat sank, the assumed mean weight for similar boats was changed from 137 lb. to 170 lb. a. Assume that a similar boat is loaded with 50 passengers, and assume that the weights of people are normally distributed with a mean of 174.1 lb. and a standard deviation of 35.1 lb. Find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than 137 lb. b. The boat was later rated to carry only 15 passengers, and the load limit was changed to 2,550 lb. Find the probability that the boat is overloaded because the mean weight of the passengers is greater than 170 (so that their total weight is greater than the maximum capacity of 2,550 lb). c. Do the new ratings appear to be safe when the boat is loaded with 15 passengers? 20. At an airport, passengers arriving at the security checkpoint have waiting times that are uniformly distributed between 0 minutes and 10 minutes. All of the different possible waiting times are equally likely. Find the probability that a randomly selected passenger has a waiting time of at least 7 minutes.
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