Descriptive Statistics Self-Test M ATH 3 1 0 F78 1. Given the data: 5.7 3.2 4.6 5.5 6.3 4.6 5.9 6.6 6.2 6.4a. Find the mean. b. Find the median.c. Find the mode. d. Find the range. e. Estimate the standard deviation. 2. For this data (lif espans, in years, of eighteen tort oises of species “ G” ):138 142 144 138 99 114 162 134 94110 148 138 148 149 145 169 140 136a. Draw a stem-and-leaf diagram for the data, using classes as indicated in part b.b. Classify the data, using intervals of w idth 10, in such a w ay that 99.5 w ill be one of the boundaries.c. Draw the histogram for the classified data.d. Find the median using the raw data. Optional: Use the histogram to estimate the median.e. Find the mean of the raw dataf. Find the interquartile range.g. Draw a box plot of the data. 3. Give an example of tw o sets of data w ith the same mean but different standard deviations.Given the data: 25 27 28 32 has mean 28 and standard deviation Ñ 2.6*(h-o had 8.67, needs square root),find data w ith a mean of 13 and the same standard deviation. (* s = 2.94) 4. a. If every value in a set of data is tripled, how are the mean and standard deviation affected?b. If ten is added to every value in a set of data, how are the mean and standard deviation affected?c. If every value in a set of data is doubled, then decreased by five, how are the mean and std. dev. affected?123d. The mean and the standard deviation of a set of " x" data (x , x , x , etc) are 42 and 10 respectively, w hatare the mean and standard deviation of the new "y" data obtained by letting each y be (x– 42)/10 for each11 22x value? (That is y = (x – 42)/10 and y = (x – 42)/10 and so on.) 5. State w hether each of the follow ing situations is possible, and, if so, give an example. Can a standard deviation of a set of data be: a. zero? b. negative? c. larger than the mean of the data? 6. a. Give an example of data w here the median is better than the mean to represent “ typical value” of a population.b. Give an example of a situation w here w e might be more interested in the mode than in the median or mean. 7. On a test, Mr Jones’ class of 40 averaged 68, & Ms. Smith’s 20 students averaged 74. Find the combined mean. 8. Tw o hundred kindergartners w ere asked their favorite color. Eighty responded " red" ; fifty " green" ; forty " blue" ;six " yellow "; and tw enty-four named other colors. Display these results in a pie chart. 9. In the follow ing pictograph, each - stands for 1000 housing units. According to the graph, how many housingunits are: a. included in the data? b. under ten years old? c. Estimate the median age of the housing units.Ages of Housing Units Age (yrs) Number of houses Scores of 250 students in BVUSD on the NMAT 0 - 9 -------------- 10-19 ------ • •20-49 ------- 76 82 86 93 9750+ ------ 10. Using the box plot above right, find the range and the interquartile range. How many of the BVUSD students scored betw een 82 and 93 on the NMAT?11. Jonathin earned 85 points on the first tw o tests, and 68 on the third. What score does he need on the upcomingfourth test so that his average on the tests w ill be at least a B (80 and above)? Should he study his head off, inan effort to earn an A ?Susan has an average of 86 for her first three tests. With a score of 100 on the fourth test, w hat w ill her average be?12. The distribution at right appears as though it could have a standard deviation close to: a. – 20 b. 0 c. 20 d. 40 e. 60 – 20 – 10 0 10 20 30 40rv1stat_xx rev s8 * * * SEE ALSO THE PRACTICE FINA L SECTION B. * * *Descriptive Statistics Self-Test Answ ers F6r8 1. a. 5.5 b. 5.8 c. 4.6 d. 3.4 e. F Ñ 1.05 ( s = {(2.3 + .9 + @@@ + 1.1 )/9} Ñ 1.11 )2 2 2 1/2 2a. Lifespans of 18 2b. 18 tortoises of species G 2d. M edian from the raw data: 139 (average 1 3 8 & 140)species G tortoises Lifespan (yrs.) f 16 29 89.5- 99.5 2 2e. raw : (94+ 99+ 110+ 114+ ... + 169)/18 = 2448/18 = 136 15 99.5-109.5 0 (If you did not have the raw data, you could estimate mean from histogram:14 0245889 109.5-119.5 2 94.5x2 + 104.5x0 + 114.5x2 + . . . + 164.5x2)/18 . 135.113 46888 119.5-129.5 0 The small difference in the result is due to the inaccuracies12 129.5-139.5 5 introduced by using the mid-values for each class11 04 Legend: 139.5-149.5 7 in place of the actual data values.) 10 9 49 149.5-159.5 0 9 49 represents 159.5-169.5 2312f. IQR = Q – Q = 148 – 134 = 14 94 yrs & 99 yrs. 1½ IQRs = 21 13 4 – 21 = 11 3 ... making 94 , 99 & 11 0 all outliers. Lifespans of 18 G tortoises, in years: t t t C C 2g. 94 99 110 114 134 139 148 169 3. Data set: 10,20,30 has mean 20, F Ñ 2.6 (s Ñ 3.1). Data set: 19,20,21 has mean 20, F Ñ.8 (s = 1).Dat a set : 10 , 1 2 , 13 , 1 7 w ill have m ean 13 (15 less than 28) and st andard deviation unchanged! 4. a. Values AND distances are tripled! so mean and s are both tripled. b. Values increase, distances betw een values do not change. Std Dev is unchanged, mean increases by 10. c. Mean is doubled, less five; s doubles. d. 0 & 1. 5. a. Yes. {4, 4, 4, 4, 4} b. No. ( q% $0) c. Yes, they are independent {-20, 0, 20} 6a. The median is a better indicator of a " typical" value w hen some extremely high or extremely low values areinvolved; for instance, in a class of tw enty students, most receive an allow ance betw een $1 and $5 per w eek,but one child receives $150/w eek— that one high value w ould raise the mean from somew here betw een $1 and$5 to betw een $8 and $12, yet only one child receives over $5. (More discussion: see DSB-6.) 6b. I am going to manufacture hats in one size only. I w ant to choose the size that w ill fit the greatest number ofw omen. Suppose, for example, 45% of w omen w ear size " S" , 30% " M " , 20% " L" , and 5% " XL" . Bymanufacturing size " S" , my hats w ill fit 4 5% of w omen. …