Extra exercise 15 ----------------- (continuation of Supplementary Exercises 6.95 and 7.64 for IPS7e; the data are in ex06_095.mtw/csv) Data: 12 readings of home radon detectors when exposed to 105 picocuries per liter of radon; the purpose being to examine the accuracy of the detectors. Model: a simple random sample (i.i.d. sample) from a distribution with unknown mean, median and standard deviation. First, we describe the distribution briefly. Estimation: sample mean = 104.1, sample median = 102.8, sample standard deviation = 9.40 The distribution could be examined using a dotplot and a stemplot. Also, descriptive statistics for skewness and kurtosis could be computed, and a normal plot and/or a normality test could be used to evaluate whether the data seem to follow a normal distribution. The distribution is somewhat right-skewed (skewness=0.85), but the normality test is far from significant. There are no obvious outliers (and no potential outliers indicated on the boxplot). As discussed in the solution for 7.64, it is not totally obvious that the t-distribution methods apply. Therefore a nonparametric analysis is of interest. Considering that the strongest apparent deviation from a normal distribution is the skewness, it would seem inconsistent to choose a nonparametric method that assumes symmetry (i.e., the Wilcoxon signed rank test). Therefore, the most obvious choice here is to use a sign test, in the following setup: H0: median=105 versus Ha: median<>105 The sign test is based the observed 4 out of 12 values above 105, and this is far from significance against H0 (P=0.39). There is no evidence that the median could not be 105. Similarly, the Wilcoxon signed rank test gives P=0.56, and the same conclusion. Also the t-test previously calculated was nonsignificant (P=0.76). Note that the t-test tests the hypothesis that the mean equals 105 but if we assume a normal distribution for the radon readings, the mean and median are the same. The bottom line is no matter how we approach the analysis, there is indication that the radon detectors are systematically off the true value of 105. --- Minitab commands and listing: MTB > WOpen "H:\VHM\VHM801\Datasets\Minitab\Chapter 6\ex06_095.mtw". Retrieving worksheet from file: ‘H:\VHM\VHM801\Datasets\Minitab\Chapter 6\ex06_095.mtw’ Worksheet was saved on 02/10/2014 MTB > Describe 'radon'. Descriptive Statistics: radon Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3 Maximum radon 12 0 104.13 2.71 9.40 91.90 96.90 102.75 109.90 122.30 MTB > GSummary 'radon'. Summary Report for radon MTB > Stem-and-Leaf 'radon'; SUBC> Trim. Stem-and-Leaf Display: radon Stem-and-leaf of radon N = 12 Leaf Unit = 1.0 1 9 1 5 9 5679 (3) 10 134 4 10 5 3 11 1 2 11 9 1 12 2 MTB > STest 105 'radon'; SUBC> Alternative 0. Sign Test for Median: radon Sign test of median = 105.0 versus not = 105.0 N Below Equal Above P Median radon 12 8 0 4 0.3877 102.8 MTB > WTest 105 'radon'; SUBC> Alternative 0. Wilcoxon Signed Rank Test: radon Test of median = 105.0 versus median not = 105.0 N for Wilcoxon Estimated N Test Statistic P Median radon 12 12 31.0 0.556 103.2 MTB > Onet 'radon'; SUBC> Test 105. One-Sample T: radon Test of mu = 105 vs not = 105 Variable N Mean StDev SE Mean 95% CI T P radon 12 104.13 9.40 2.71 (98.16, 110.10) -0.32 0.755 Addition: "manual" calculation of rank test statistic ----------------------------------------------------- We do the calculations in Minitab. First calculate the differences to the hypothesized median, then compute their absolute values, and finally compute the ranks. After the data have been sorted, we can directly read off the ranks in "positive" part of the distribution (above 105) as 2, 6, 11 and 12, with a sum of 31. MTB > Let 'diff105' = 'radon'-105 MTB > Name C4 'absdiff105' MTB > Let 'absdiff105' = abs('diff105') MTB > Name c5 "ranks" MTB > Rank 'absdiff105' 'ranks'. MTB > Sort 'id'-'ranks' 'id'-'ranks'; SUBC> By 'radon'. MTB > Print 'id'-'ranks'. Data Display Row id radon diff105 absdiff105 ranks 1 1 91.9 -13.1 13.1 10 2 6 95.0 -10.0 10.0 9 3 9 96.6 -8.4 8.4 8 4 2 97.8 -7.2 7.2 7 5 8 99.6 -5.4 5.4 5 6 12 101.7 -3.3 3.3 4 7 7 103.8 -1.2 1.2 3 8 11 104.8 -0.2 0.2 1 9 5 105.4 0.4 0.4 2 10 3 111.4 6.4 6.4 6 11 10 119.3 14.3 14.3 11 12 4 122.3 17.3 17.3 12