MATH 106 Week 7 LEO Discussion Topic Exercises Revised: 25 June 2016 NORMAL DISTRIBUTION (Online Statistics Education: A Multimedia Course of Study: Chapter 7, Sections: \"Areas of Normal Distribution\" ; \"Standard Normal\" ; \"Normal Approximation to Binomial\") AREAS OF NORMAL DISTRIBUTION 1. For a normal distribution with mean of 100 and a standard deviation of 12, find the number of standard deviations the raw score 76 is from the mean (Give answer as a positive value rounded to nearest hundredth). 2. For a normal distribution with mean of 120 and a standard deviation of 15, find the number of standard deviations the raw score 144 is from the mean (Give answer as a positive value rounded to nearest hundredth). 3. For a normal distribution with mean of 80 and a standard deviation of 8, find the number of standard deviations the raw score 67 is from the mean (Give answer as a positive value rounded to nearest hundredth). 4. For a normal distribution with mean of 62 and a standard deviation of 6, find the number of standard deviations the raw score 47 is from the mean (Give answer as a positive value rounded to nearest hundredth). 5. A normal distribution has a mean of 40 and a standard deviation of 5. 68% of the distribution can be found between what two numbers? 6. A normal distribution has a mean of 20 and a standard deviation of 3. Approximately 95% of the distribution can be found between what two numbers? 7. A normal distribution has a mean of 5 and a standard deviation of 2. What proportion of the distribution is above 3? Use either the Area Under Normal Distribution Calculator or Z - table link (round answer to nearest thousandth). 8. A normal distribution has a mean of 12 and a standard deviation of 3. What proportion of the distribution is below 7? Use either the Area Under Normal Distribution Calculator or Z - table link (round answer to nearest thousandth). 9. A normal distribution has a mean of 120 and a variance of 100. 35% of the area is below what number? Use either the Inverse Normal Distribution Area Calculator or or Z - table link (round answer to nearest thousandth). 10. A normal distribution of test scores has a mean of 38 and a standard deviation of 6. Everyone scoring at or above the 80th percentile gets placed in an advanced class. What is the cutoff score to get into the class? Use either the Inverse Normal Distribution Area Calculator or or Z - table link (round answer to nearest thousandth). 1 MATH 106 Week 7 LEO Discussion Topic Exercises Revised: 25 June 2016 11. A normal distribution of test scores has a mean of 38 and a standard deviation of 6. What percent of the students scored between 30 and 45? Use either the Area Under Normal Distribution Calculator or Z - table link (round answer to nearest tenth of a percent). STANDARD NORMAL 12. A normal distribution has a mean of 16 and a standard deviation of 6. What is the Z score that corresponds with 25? 13. A normal distribution has a mean of 100 and a standard deviation of 10. What is the Z score that corresponds with 122? 14. A normal distribution has a mean of 84 and a standard deviation of 7. What is the Z score that corresponds with 95? 15. A normal distribution has a mean of 75 and a standard deviation of 15. What is the raw score that corresponds with a Z score of 1.6? 16. A normal distribution has a mean of 200 and a standard deviation of 25. What is the raw score that corresponds with a Z score of - 1.5? 17. A normal distribution has a mean of 1200 and a standard deviation of 150. What is the raw score that corresponds with a Z score of - 0.5? 18. A normal distribution has a mean of 18 and a standard deviation of 5. Use either the Area Under Normal Distribution Calculator or Z - table link to determine the proportion of the scores (area) below 6. 19. Use either the Area Under Normal Distribution Calculator or Z - table link to find the area under the standard normal curve from Z = 0 to Z = 1.4. 20. Use either the Area Under Normal Distribution Calculator or Z - table link to find the area under the standard normal curve from Z = - 2.5 to Z = 0. 21. Use either the Area Under Normal Distribution Calculator or Z - table link to find the area under the standard normal curve from Z = 0 to Z = 0.67. 22. Use either the Area Under Normal Distribution Calculator or Z - table link to find the area under the standard normal curve from Z = - 1.2 to Z = 0.75. 23. Use either the Area Under Normal Distribution Calculator or Z - table link to find the area under the standard normal curve from Z = 0.3 to Z = 2.6. 2 MATH 106 Week 7 LEO Discussion Topic Exercises Revised: 25 June 2016 24. A normal distribution has a mean of 100 and a standard deviation of 10. What is the Z score that corresponds with 122? Use either the Area Under Normal Distribution Calculator or Z - table link to find the area under the standard normal curve from the mean to raw score 122. 25. A normal distribution has a mean of 50 and a standard deviation of 5. Use either the Area Under Normal Distribution Calculator or Z - table link Use the table found at the link Z - table to find the area under the standard normal curve between raw score of 39 and raw score 59. 26. \"Cut-It-Up\" brand chain saws have an average service life of 120 months with a standard deviation of 15 months. Assuming a normal distribution, find the probability that a \"Cut-It-Up\" chain saw will last for more than 140 months. Compute Z score and use either the Area Under Normal Distribution Calculator or Z - table link to compute the corresponding probability. 27. Sales of \"Cut-It-Up\" brand chain saws average $250,000 per year with a standard deviation of $20,000. Assuming a normal distribution, find the probability that \"Cut-It-Up\" chain saw annual sales will be more than $300,000. Compute Z score and use either the Area Under Normal Distribution Calculator or Z - table link to compute the corresponding probability. 28. The mean time for ice cream to freeze is 210 minutes, with a standard deviation of 15 minutes. Assuming a normal distribution, find the probability that ice cream will take less than 190 minutes to freeze. Compute Z score and use either the Area Under Normal Distribution Calculator or Z - table link to compute the corresponding probability. 29. You're the production manager at world-famous Jack's Breads. The average length of a loaf of \"Jack's Bread\" is 11.0 inches, with a standard deviation of 0.5 inches. Assuming a normal distribution, find the probability that a loaf of bread off the production line will be 11.8 inches or longer. Compute Z score and use either the Area Under Normal Distribution Calculator or Z - table link to compute the corresponding probability. 30. If 120 measurements of the distance between home and work are randomly chosen from a normal distribution with mean of 15 and standard deviation of 4, how many raw measurements x would be expected to fall between (and including) 10 and 19? Use either the Area Under Normal Distribution Calculator or Z - table link to compute the corresponding area under the curve to determine the number of measurements. 31. The average healing time from a particular inpatient surgical procedure is 200 hours, with a standard deviation of 20 hours. What percentage of the people having this surgical procedure will heal in 170 hours or less? Use either the Area Under Normal Distribution Calculator or Z - table link to compute the corresponding area under the curve to determine the percentage. 3 MATH 106 Week 7 LEO Discussion Topic Exercises Revised: 25 June 2016 NORMAL APPROXIMATION TO BINOMIAL 32. Suppose you have a normal distribution with a mean of 6 and a standard deviation of 1. What is the probability of getting a Z score of exactly 1.2? Use either the Area Under Normal Distribution Calculator or Z - table link to compute the corresponding probability - but be careful!! 33. You decide to use the normal distribution to approximate the binomial distribution. You want to know the probability of getting exactly 6 tails out of 10 flips. Find the mean and standard deviation of the normal distribution, then compute Z score and use either the Area Under Normal Distribution Calculator or Z - table link to compute the corresponding probability. 34. You decide to use the normal distribution to approximate the binomial distribution. You want to know the probability of getting from 7 to 13 heads out of 20 flips of a fair, two - sided coin. Find the mean and standard deviation of the normal distribution, then compute Z scores and use either the Area Under Normal Distribution Calculator or Z - table link to compute the corresponding area under the curve to determine cumulative probability. 35. Is a normal distribution with the same mean and standard deviation as a binomial distribution a viable approximation to use if the binomial distribution's number of trials n = 150 and its probability of success p = 0.01? Why or why not? 4