Unit 2 Problems Set NAME: Elements of Statistics--FHSU Virtual College--Summer 2015 REMEMBER, these are assessed preparatory problems related to the content of Unit 2. The Unit 2 Exam will consist of similar types of problems, but not exactly the same. Thus, make sure you are thinking about the concepts and procedures you studied in this unit versus simply \"copying\" the process of an example problem. Also, take time to examine the complete objective list in the Unit 2 Review document. Listed out to the left of the spreadsheet are text chapter separators if you find yourself needing some direction to a related resource. All answers should be calculated, as needed, within this Excel sheet, and final concluding answers given directly below or to the right of the problem. Please make your answers easily found--for example use a different color or type of font. No numerical answer resulting from a calculation will be accepted unless the process is performed in Excel and formulas/calculations used are evident when the cell is selected. Type your name at the top, complete and return this file saved as "yournameUnit2ProblemSet" through the Exam Prep region in Blackboard (from the same location that you downloaded this file.) This problem set is due no later than the date set by your instructor for this course. Your instructor will grade and return it to you with feedback to assist you in preparing for the Unit 2 Exam. Problems related to text's Chapter 5 (5-1 to 5-4) 1. Identify each of the random variables described below as discrete or continuous a. The number of houses in a randomly chosen neighborhood in Chicago. b. The height of a randomly selected door in Rarick Hall (on the FHSU campus). c. The high temperature in degrees Fahrenheit on July 4th in a randomly selected city in the US. d. The cost to make that randomly selected movie. 2. The incomplete probability distribution table at the right is of the discrete random variable x representing the number of dogs owned by each family in a small town. Answer the following: (a) Determine the value that is missing in the table. (Hint: what are the requirments for a probability distribution?) (b) Find the probability that x is at least 3 , that is find P(x 3). x 0 1 2 3 4 P(X=x) 0.433 0.289 0.115 x P(x) 0.012 (c) Find P(x 2). Describe what the resulting value represents within the given context. (d) Find the mean (expected value) and standard deviation of this probability distribution. 3. (a) What is meant by the term "expected value"? (b) Suppose the expected value of a slot machine (game of chance) in Las Vegas is a negative 5 cents. Does this mean that every time Alice plays this game, she will lose 5? Explain your answer briefly. 4. List the four requirements needed for an experiment/procedure to be considered a binomial distribution. 5. A quality control expert at a large factory estimates that 5% of all the batteries produced at the factory are defective. (The other 95% are not defective.) Six batteries are randomly selected and then each is tested to see if it's defective. Use this situation to answer the parts below. (a) Recognizing that this is a binomial situation, give the meaning S and F in this context. That is, define what you will classify as a "success" and what you will classify as a "failure" when one battery is selected and tested for defectiveness. S= F= (b) Next, give the values of n, p, and q. n= p= q= (c) Construct the complete binomial probability distribution for this situation in a table out to the right. (d) Using your table, find the probability that exactly one of six randomly selected batteries is defective. (e) Find the probability that at least three of the six batteries selected are defective. (f) Find the probability that less than two are defective. (g) Find the mean and standard deviation of this binomial probability distribution. (h) By writing a sentence, interpret the meaning of the mean value found in (g) as tied to the context of defectiveness among six randomly chosen batteries. (i) Is it unusual to have all six of the batteries selected be defective? Briefly explain your answer giving supporting numerical evidence. Problems related to text's Chapter 6a (6-1 to 6-3) 6. What is a normal distribution? What is a standard normal distribution? Briefly compare and contrast these two statistical terms. 7. With regards to a standard normal distribution complete the following: (a) Find P(z < 0), the proportion of the standard normal distribution below the z-score of 0. (b) Find P(z > 1.5), the percentage of the standard normal distribution above the z-score of 1.5. (c) Find P(-1.2 < z < 2.2). (d) Find P( z < 0.8). (e) Find the z-score that separates the lower 90% of standarized scores from the top 10% . . . that is find the zscore corresponding to P90, the 90th percentile value. 8. If the results on a nationally administered introductory statistics exam is normally distributed with a mean of 100 points and a standard deviation of 8 points, determine the following: (a) Describe the graph of this distribution (if you can do so, produce an electronic sketch of the graph to the right, otherwise adequately describe the distribution graph through its shape and horizontal scale values.) (b) Find the z-score for a single exam that had 90 points. Then find the z-score for one with 118 points. (c) If x represents a possible point-score on the exam, find P(x < 90). (d) Find P(90 < x < 118) and give an interpretation of this value. (e) What is the minimum number of points one must score on this exam if they want to be in the top 30% of all the scores? Problems related primarily to text's Chapter 6b (6-4 to 6-5) 9. Explain what a \"sampling distribution of sample means\" is. Be specific! 10. Fill in the blanks in the statements below. The Central Limit Theorem states that as the sample size increases, the distribution of all the possible sample means (that is the sampling distribution of sample means) approaches a ______________ distribution. The mean of the sampling distribution will be ______________ as/than the mean of the population and the standard deviation of the sampling distribution will be ______________ as/than the standard deviation of the population. (Fill in the first blank with one word. Fill each of the other blanks with a "comparison" word (or phrase). 11. The walking gait of an adult male giraffe is normally distributed with a mean of 8 feet and a standard deviation of 1.2 feet. Complete the following. (a) Describe the shape and horizontal scaling on the graph of the distribution for the population of all adult male walking gaits (hereafter referred to simply as gaits). (b) Find the probability that the gait of a randomly selected adult male giraffe will be less than 7 feet---that is, find P(x < 7). Based upon your result, state whether or not it is unusual to randomly select an adult male giraffe whose gait is less than 7 (and explain why you chose "unusual" or "not unusual" as your answer). (c) Suppose all possible samples of size 36, taken from the population of all adult male giraffe gaits, are drawn and the mean is found for each resulting sample. Describe the shape and scaling on the graph of the resulting sampling distribution for the sample mean values. Hint: Apply the Central Limit Theorem! (d) Find the probability that the mean gait of a randomly selected sample of 36 adult male giraffes will be less than 7.2 feet---that is, find P(x-bar < 7.2). Based upon your result, state whether or not it is unusual to randomly select a sample of 36 adult male giraffes whose mean gait is less than 7.2 (and explain why you chose "unusual" or "not unusual" as your answer). (e) Find the probability that the mean from a sample of 36 gaits will be between 8.0 and 8.25 feet. Problems related to text's Chapter 7a (7-1 to 7-3) 12. Suppose that a 90% confidence interval for a population mean is given to be from 550 to 650. The 90% value in this statement indicates that: (choose the one best answer) A. P ( 550 < < 650 ) = 0.90 B. There is a 10% chance that falls between 550 and 650. C. 13. 14. When the statistical techniques in this unit are applied to build a 90% confidence interval around the mean of any randomly collected sample, we can only expect 90% of such calculated intervals to contain the true value of . D. There is only a 10% chance that the collected sample mean will be between 550 and 650 . Suppose that a 95% confidence interval for a population mean is constructed from a sample, resulting in the interval from 550 to 650. What was the value of the sample mean that was collected? What was the size of the margin of error? A sample of size n = 40 is taken. What is the appropriate critical t-value for each of the following confidence levels? (a) 90% confidence level: (b) 96% confidence level: 15. Does a confidence interval get wider or narrower if: (a) the percent of desired confidence increases from 90% to 95%? (b) the size of the sample used to produce the confidence interval is decreased? On problems 16-19, show your work to the right of each problem. Use Excel commands as appropriate. 16. A graduate student wishes to know the proportion of U.S. adults who speak two or more languages. He surveys 565 U.S. adults and finds that 113 speak two or more languages. Construct a 99% confidence interval to estimate the proportion of all U.S. adults that speak two or more languages. 17. The Montana State Education Commission wants to estimate the percentage of tenth-grade students who read at or below the eighth-grade level. How large a sample should be selected in order to estimate this proportion with a 90% confidence level and a margin of error of at most 4%? (Assume no preliminary estimate of p-hat is available.) 18. A toy company wants to know the average number of new toys bought for children each year. Marketers at the toy company collect data from the parents of 50 children. The sample mean is 4.2 toys with a sample standard deviation of 1.5 toys. Construct a 95% confidence interval to estimate the population mean. 19. Redo #18, only this time assume the value 1.5 is the population standard deviation. 20. For a psychology experiment, Emma is assigned the task of finding the average number of hours U.S. college students sleep each night. Her results must be at the 90% level of confidence with a maximum margin of error of 0.25 hours. Assuming the standard deviation is 1.3 hours (based on a previous, reliable study), how many students must Emma survey