1.) A study of sizes of metastatic carcinoid tumors in the heart was conducted by U. Pandya et al. and published as the article, \"Metastatic Carcinoid Tumor to the Heart: Echocardiographic-Pathologic Study of 11 Patients\" (Journal of the American College of Cardiology, Vol. 40, pp. 1328-1332). Based on that study, we assume that lengths of metastatic carcinoid tumors in the heart are normally distributed with mean 1.8 cm and standard deviation 0.5 cm. a.) Find the percentage of metastatic carcinoid tumors in the heart that are between 1.5 cm and 2.5 cm long. b.) Find the percentage of metastatic carcinoid tumors in the heart that exceed 3.2 cm in length. c.) How large could a tumor grow before a researcher might consider it to be an outlier and would want to double check the measurement. d.) Suppose a tumor had a length which was in the lower 15% of all metastatic carcinoid tumors. At most, how large could it have been? e.) The middle 95.44% of all metastatic carcinoid tumors have lengths between ______ and _____. f.) The middle 76% of all metastatic carcinoid tumors have lengths between ______ and _____. 2.) A random sample of 20 pregnancies was taken and the gestation length (in days) is given below: 280 292 289 249 270 272 203 268 275 258 294 286 255 280 275 264 a.) Create a normal probability plot. List all your data points in the table below. x y b.) Which data value(s) is/are considered outliers? Why? 268 256 287 279 c.) Remove the outlier(s) from the data set and create a new normal probability plot. List all your data points in the table below. Hint: all your y-coordinates should now be different values than they were previously, and you will have at least one empty set of values in the table, as you have removed at least one outlier. x y d.) Now, is the data approximately normally distributed? Why? Suppose it is known that the population mean gestation period is 266 days with a standard deviation of 16. e.) Was/were your outlier(s) more than three standard deviations away from the mean? If so, exactly how many standard deviations away from the mean was/were the outlier(s)? f.) What percent of the population should have a pregnancy that lasts less than 282 days? g.) What percent of the population would you expect to have a pregnancy that lasts longer than 230 days? h.) How do your answers to (f) and (g) compare with the sample results? 3.) Suppose the data in the first column represents 18 scores from the first statistics exam. (I've filled out the table for you to make things easier.) a.) Given this table, create a probability plot. Exam Normal xy Scores (x) Score (y) 59 -1.82 -107.38 59 -1.35 -79.65 60 -1.06 -63.6 65 -0.84 -54.6 70 -0.66 -46.2 75 -0.5 -37.5 83 -0.35 -29.05 88 -0.21 -18.48 92 -0.07 -6.44 93 0.07 6.51 95 0.21 19.95 95 0.35 33.25 97 0.5 48.5 97 0.66 64.02 98 0.84 82.32 99 1.06 104.94 100 1.35 135 100 1.82 182 1,525 0 233.59 x2 3481 3481 3600 4225 4900 5625 6889 7744 8464 8649 9025 9025 9409 9409 9604 9801 10000 10000 133,331 y2 3.3124 1.8225 1.1236 0.7056 0.4356 0.25 0.1225 0.0441 0.0049 0.0049 0.0441 0.1225 0.25 0.4356 0.7056 1.1236 1.8225 3.3124 15.6424 b.) Are the exam scores approximately normal? Why or why not? c.) If you were to form a regression line from your normal probability plot, what would be the value of the correlation coefficient? (Hint: You'll want to compute S xx , S xy , and S yy first.) 4.) Below is a list of the number of homeruns hit by each baseball team in 2000. (There are 30 teams, so I've given you the normal probability plot numbers from the appendix of your textbook, and to save time, I've filled in the chart for you...) Total HR hit Normal TEAM xy x2 y2 by team (x) Score (y) -2.04 4.1616 SEA 101 -206.04 10201 -1.61 2.5921 HOU 108 -173.88 11664 -1.36 1.8496 OAK 109 -148.24 11881 -1.17 1.3689 LAD 120 -140.4 14400 -1.02 1.0404 KC 121 -123.42 14641 -0.89 0.7921 PIT 126 -112.14 15876 -0.77 0.5929 CLE 128 -98.56 16384 -0.67 0.4489 NYM 128 -85.76 16384 -0.57 0.3249 SD 132 -75.24 17424 -0.47 0.2209 BAL 133 -62.51 17689 -0.38 0.1444 ATL 139 -52.82 19321 -0.29 0.0841 MIN 142 -41.18 20164 -0.21 0.0441 CHI 149 -31.29 22201 -0.12 0.0144 WASH 149 -17.88 22201 -0.04 0.0016 STL 150 -6 22500 0.04 0.0016 DET 152 6.08 23104 0.12 0.0144 FL 152 18.24 23104 0.21 0.0441 LAA 155 32.55 24025 0.29 0.0841 TB 160 46.4 25600 0.38 0.1444 TEX 162 61.56 26244 0.47 0.2209 SF 162 76.14 26244 0.57 0.3249 PHI 166 94.62 27556 0.67 0.4489 COL 173 115.91 29929 0.77 0.5929 CHI 177 136.29 31329 0.89 0.7921 ARI 180 160.2 32400 1.02 1.0404 MIL 182 185.64 33124 1.17 1.3689 CIN 188 219.96 35344 1.36 1.8496 NYY 201 273.36 40401 1.61 2.5921 BOS 211 339.71 44521 2.04 4.1616 TOR 257 524.28 66049 TOTALS 4,613 0 27.3618 915.58 741,905 a.) Find the following values: S xx S xy S yy b.) Create a normal probability plot. c.) Does the data appear to be approximately normal? Explain your answer. d.) If you were to find the regression line for your plot, what would the value of r be? e.) Compare your answers to 3c and 4d...For which set of data was r larger? Why would that make sense? 5.) Compute the following values: a.) P ( z > 2.37) b.) P ( z 1.53) c.) P (- 2.82 < z <- 0.18) d.) z.03 e.) z.005 f.) Find a if P ( z