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Practice Test Chapter 6 (Normal Probability Distributions)

This document provides a study guide for a statistics exam, covering normal probability distributions, sample sizes, and estimation principles. It includes exercises on calculating probabilities, confidence intervals, and the use of t-distribution, as well as applications to real-life scenarios like IQ scores, boat weight limits, and standardized tests. The content is organized by chapters with specific questions aimed at reinforcing understanding of statistical concepts.

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1 Statistics, Sample Test (Exam Review) Module 3: Chapters 6 & 7 Review Chapter 6: Normal Probability Distributions Chapter 7: Estimates & Sample Sizes Chapter 6: Normal Probability Distributions 1. Answer the following: a. What percent of normally distributed data value lie within 2 standard deviations to either side of the mean? b. For a continuous random variable, the probability of a single value of x is always _ c. The parameters of normal distribution are ____________ d. What are the values of mean and the standard deviation for the standard normal distribution? 2. Answer the following: a. Find the value of z for a standard normal distribution, such that the area in the left tail is 0.05 b. Find the value of z for a standard normal distribution, such that the area in the right tail is 0.1. c. Find the following probability: P(z > -0.82) d. The z-value for μ of normal distribution curve is always ________ e. Usually, the normal distribution is used as an approximation to binomial distribution when _________ 3. Find the area of the shaded regions. a. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1. b. The graph depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15. a. b.
2 4. 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. a. Find the probability that a given score is between −2.13 and 3.77 and draw a sketch of the region. b. Draw a graph and find P19, the 19th percentile. This is the bone density score separating the bottom 19% from the top 81 %. 5. 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 6. 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%? 7. 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.
3 8. 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? 9. 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? 10. 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? 11. 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. 12. Information regarding standardized tests used for college admittance.
4 Scores on the SAT test are normally distributed with a mean of 984 and a standard deviation of 204. Scores on the ACT test are normally distributed with a mean of 20.7 and a standard deviation of 4.3. It is assumed that the two tests measure the same aptitude, but use different scales. a. If a student gets an SAT score that is the 65-percentile, find the actual SAT score (Whole number). b. What would be the equivalent ACT score for this student (1 decimal)? c. If a student gets an SAT score of 1494, find the equivalent ACT score (1 decimal). Statistics, Sample Test (Exam Review) Module 3: Chapters 6 & 7 Review Chapter 7: Estimates & Sample Sizes 1. Answer the following: a. True or False: The confidence interval estimate of the population mean is constructed around the sample mean. ________ b. Estimation means assigning values to a population parameter based on the value of a ________ c. The confidence level is denoted by ________ d. The maximum error (margin of error) of the estimate for μ (based on known σ) is: e. The parameter(s) of t distribution is (are) _________ 2. Answer the following: a. What criteria are required to apply the t distribution to make a confidence interval for μ? b. Find the values for t for a t-distribution with sample size of 20 and a confidence level of 95%. c. Find the values for t for a t-distribution with sample size of 30 and a confidence level of 90%. 3. Find the critical chi-square value a. for 20 degrees of freedom when the area to the left is 0.01
5 b. Find the critical chi-square value for n=30 when the area to the right is 0.9 degrees of freedom: 29 4. A statistician is interested in estimating at a 95% confidence level the mean number of houses sold per month by all real estate agents in a large city. It is known that the population standard deviation is 1.9. a. How large a sample should be taken so that the estimate is within 0.65 of the population mean? b. Find this 95% confidence interval, if the mean for such sample size is 3.4 5. A random sample of 20 female members of a health club showed that they spend on average 4.5 hours per week doing physical exercise with standard deviation of 0.75 hours (assume the time spent on exercise for all female members is approximately normally distributed) a. What is the value of the point estimate of the population mean? b. Construct the 98% confidence interval for the mean time spent on physical exercise of all such females. 6. Out of 500 randomly selected adults, 300 said they were in favor of the death penalty for a person convicted of murder. a. What is the value of point estimate of the population proportion? b. What is the 95% confidence interval for the population proportion? 7. How large a sample is needed to be 99% confident that the margin of error is 0.025, for percentage of golfers that are left-handed, if we a. Assume that 15% of the golfers are left-handed, based on a previous study. b. Assume that we have no prior information suggesting a possible value for the sample proportion. 8. Given: 30, 80.5, 4.6 n x s = = = (assume normal population) a. Find 98% confidence interval for population variance. b. Find 98% confidence interval for population standard deviation.