Unit 3: Statistics Lab STUDENT HANDOUT Student Name: Learning Outcomes: 1. Describe data values in real world contexts using appropriate measures of central tendency and spread. 2. Given a mean and standard deviation of a distribution determine the zscore and percentile of a given data point and communicate this information in a meaningful sentence. Introduction: Many things in the real world are distributed normally, including height, IQ, and test scores. In this lab, we will explore normally distributed data and analyze the distribution through standard deviation, zscores, and percentiles. Directions: Show all of your work, where necessary. PART I 1) In a normal distribution, what percent of the values lie: a. below the mean? ________________ b. above the mean? ________________ c. within one standard deviation of the mean? ________________ d. within two standard deviations of the mean? ________________ e. within three standard deviations of the mean? ________________ 2) 2000 freshmen at State University took a biology test. The scores were distributed normally with a mean of 70 and a standard deviation of 5. Label the mean and three standard deviations from the mean. 3) 4) 5) a) What percentage of score are between scores 65 and 75? 6) 7) b) What percentage of score are between scores 60 and 70? 8) c) What percentage of score are between scores 65 and 85? 9) 10) 11) d) What percentage of scores is less than a score of 55? 12) 13) 14) e) What percentage of scores is greater than a score of 80? 15) 16) 17) f) Approximately how many biology students scored between 60 and 70? 18) 19) 20) g) Approximately how many biology students scored between 55 and 60? 21) 22) 23) 24) 25) Given a class of 26 students, here are the scores for a recent test in MAT 152 class: 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 90 90 95 100 80 80 75 80 70 60 39) 40) 41) 42) 43) 44) 45) 46) 47) 48) 95 100 100 100 75 80 90 90 90 70 52) 53) Answer the following questions regarding this set of data. a) mean d) standard deviation b) median 36) 70 49) 90 37) 80 50) 90 38) 85 51) 85 c) mode e) variance f) How many scores are within one standard deviation of the mean? (Show your work) g) h) i) j) How many scores are within two standard deviations of the mean? (Show your work) k) l) m) 54) A normal distribution of scores has a standard deviation of 10. Find the z-scores corresponding to each of the following values: a) A score of 60, where the mean score of the sample data values is 40. n) o) p) b) A score that is 30 points below the mean. q) r) s) c) A score of 80, where the mean score of the sample data values is 30. t) 55) IQ scores have a mean of 100 and a standard deviation of 16. Albert Einstein reportedly had an IQ of 160. a) What is the difference between Einstein's IQ and the mean? u) v) b) How many standard deviations is that? w) x) y) z) c) Convert Einstein's IQ score to a z-score. aa) ab) ac) ad) d) If we consider \"usual IQ scores to be those that convert z-scores between -2 and 2, is Einstein's IQ usual or unusual? Explain. ae) af) ag) ah) ai) aj) ak) 56) Three students take equivalent stress tests. Which is the highest relative score (meaning which has the largest z score value)? al) a. A score of 144 on a test with a mean of 128 and a standard deviation of 34. am) an) ao) b. A score of 90 on a test with a mean of 86 and a standard deviation of 18. ap) aq) ar) c. A score of 18 on a test with a mean of 15 and a standard deviation of 5. as) at) au) av) A percentile is a number that tells us the percent of a data set that lies at or below a given value. Percentiles are mostly used when working with large data sets (data sets with over 100 values) in order to give particular values a standardized ranking compared to others. That is why they are particularly useful on large standardized tests where many thousands of scores are recorded. aw) 57) In a race with 350 competitors, a. In what percentile did the 40th placed competitor rank? ax) ay) az) b. If you finished in the 65th percentile, how many competitors did you beat? ba) bb) bc) c. If you finished in the 80th percentile, how many competitors finished before you? bd) 58) Madaline and Preeda are trying to determine who did better on a standardized test. When Madaline took it in the fall, her score placed her better than 240 other participants out of 300. When Preeda took it in the spring, her score placed her better than 265 other participants out of 350. Based on their percentile rank, who did better, Madaline or Preeda? (Show your work) be) bf) 59) The following box-and-whiskers diagram gives the results on a recent SAT math test. bg) a) b) c) d) e) What is the 25th percentile score? _______ What is the 50th percentile score? _______ What is the 75th percentile score? _______ What is the 100th percentile score? ______ Label the quartiles. bh) bi) bj) bk) 60) Quentin's height is between the median and the third quartile for all heights in his school. His height's percentile rank could be which of the following? a. 45th b. 79th c. 64th d. 23rd e. f. g. h. i. j. 61) 280 students took a math test at DCCC. Malik's score of 88 put him in the 92 th percentile. How many students did the same or better than Malik on this math test? a. b. c. d. e. f. g. h. i. j. k. l. m. n. o. p. q. r. s. t. u. v. w. x. y. z. aa. The 68-95-99.7 Rule for the Normal Distribution ab. 99.7% 95% 68%% % -3 1 -2 -1 2 3 ac. ad. 1) 2) 3) 4) 5) ae. af. ag. PART II ah. ai. For questions that ask you to plot or write on the graph, use the graph directly above. aj. ak. For Women: al. The distribution of heights of young women is approximately normal with a mean of 65 inches and a standard deviation of 3.5 inches. am. an. For Men: ao. The distribution of heights of young men is approximately normal with a mean of 70 inches and a standard deviation of 4 inches. ap. What is the mean for the height of your gender? Plot the mean on the graph. aq. ar. Find and plot the heights that are 1, 2, and 3 standard deviations below and above the mean. as. at. au. av. aw. ax. ay. What height is at 2? az. ba. What height is at -1? bb. bc. Find and write on the graph the percent represented under the graph between each standard deviation line. 6) 7) 8) 9) 10) 11) 12) 13) bd. be. bf. bg. What percent of persons fall below the answer in question #3? bh. bi. bj. What percent of persons are above the answer in question #4? bk. bl. bm. What percent of persons fall between #3 and #4? bn. bo. bp. bq. Women: Find and plot a height of 60 in. Men: Find and plot a height of 65 in. br. Z60 = Z65 = bs. bt. bu. bv. bw. bx. How tall would a man/woman be with a Z-score of 1.5? by. bz. ca. cb. Determine your Z-score. cc. cd. ce. cf. What is your height percentile? cg. ch. ci. cj. ck. a. Quartile? cl. cm. cn. co. cp. How many women/men are taller than you? cq. cr. cs. ct