330 Model 1 - StatTools Assignment#1 Solution Part I 2a Trading volume One Variable Summary Mean Std Dev Median Minimum Maximum Count 1st Quartile

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Unformatted text preview: StatTools Assignment #1 Solution Part I: 2a. Trading volume One Variable Summary Mean Std. Dev. Median Minimum Maximum Count 1st Quartile 3rd Quartile Interquartile Range Data Set #1 Revenue growth Data Set #1 579338.41 1496813.67 27392.00 100.00 6696185.00 22 4575.00 250350.50 245775.50 -0.0224 0.2180 -0.0006 -0.5560 0.4337 22 -0.1532 0.0798 0.2329 2b. Trading Volume: Q-Q Normal Plot of Trading volume / Data Set #1 Histogram of Tra ding volume / Da ta Set #1 3 20 18 16 2 Standardized Q-Value 1 14 Frequency 12 10 8 6 4 2 0 558107.08 1674121.25 2790135.42 3906149.58 5022163.75 6138177.92 StatTools Student Version For Academic Use Only 0 -3 -2 -1 -1 -2 0 1 2 3 -3 Z-Value Revenue Growth: Histogra m of Revenue grow th / Data Set #1 12 10 8 Frequency 6 4 2 -2 0 -0.4735 -0.3086 -0.1436 0.0213 0.1863 0.3512 -3 Z-Value Standardized Q-Value Q-Q Norma l Plot of Re ve nue grow th / Data Se t #1 3 2 1 StatTools Student Version For Academic Use Only 0 -3 -2 -1 -1 0 1 2 3 3a. The Revenue Growth data appears to be approximately Normal. This can be seen by the general "Bell Shape" of the data in the histogram, as well as the fact the the data points form an approximate 45-degree line in the Q-Q Normal Plot. Further, looking at the One Variable Summary, we can see that the Mean and Median (-0.0224 and -0.0006, respectively) are very close together, and that both the minimum and maximum and 1st and 3rd quartiles are approximately as far apart to each side of the mean/median, again, indicating that the data is approximately symmetric. [Bonus: Other ways to verify Normality might include checking the values of the quartiles. The quartiles for a Normal distribution are approximately 0.675 (= Z) standard deviations away from the mean. Using the Normal theory, the quartiles would be x (.675) s = -.0224 .675(.2180). The first quartile as calculated using Normal probabilities would be -.16955, while the sample Q1 = -.1532. The third quartile as calculated using Normal probabilities would be .12475, while the sample Q3 = .0798. The sample quartiles are similar to the expected quartiles, providing further evidence that the Revenue Growth data is approximately Normal.] The Trading Volume data, however, appears to be severely skewed to the right, with outliers. This can be seen by the long tail that drags out to the right in the histogram, and the lack of a clear 45-degree line in the Q-Q plot. Further, the mean (579,338.41) is significantly larger than the mean (27,392), again, indicating that the data is rightly skewed. A check of the quartiles could be done for the Trading Volume data also, but is really unnecessary. The skewness is very obvious from more basic information. 3b. Looking at the graphs for the Revenue growth, there don't appear to be any strong outliers. For trading volume, however, because the data doesn't follow a 45-degree line, it is clear that we are going to have several outliers on our hands. 3c. Trading Volume MIN MAX Q1 1.5 x IQR Q3 + 1.5 x IQR 100.00 6,696,185.00 (4575) 1.5(245775.5) = -364088.25 (250350.5) + 1.5(245775.5) = 619013.75 Yes, 4 high: COVD @ 2,410,204 HLIT @ 690,026 QCOM @ 6,696,185 XXIA @ 1,750,027 Revenue Growth -0.5560 0.4337 (-0.1532) 1.5(0.2329) = -0.50255 (0.0798)+ 1.5(0.2329) = .42915 Yes, 1 high: GCOM @ .4337 and 1 low: PTSC @ -.556 Outlier 3d. x-bar 1s 68% of n x-bar 2s 95% of n x-bar 3s 99.7% of n Trading Volume (579338.41) (1496813.67) captures 20 of 22, or 91% 15 of 22 (579338.41) 2(1496813.67) captures 21 of 22, or 95% 21 of 22 (579338.41) 3(1496813.67) captures 21 of 22, or 95% all 22 Revenue Growth (-0.224) 0.2180 captures 9 of 22, or 41% 15 of 22 (-0.224) 2(0.2180) captures 20 of 22, or 91% 21 of 22 (-0.224) 3(0.2180) captures 21 of 22, or 95% all 22 Both sets of data are inconsistent with the expected percentage of data points in the one standard deviation range. Ninety-one percent of the Trading Volume data values fall within 1 standard deviation of the mean as compared to the theoretical 68%. This is very consistent with the skewness seen in the graph. The results for the Revenue Growth data show a smaller (41%) than expected percentage (68%) of points in the 1 standard deviation range. Some difference from the theoretical should always be expected, and especially so when the sample size is small. While the 1.5 IQR rule identified two possible (moderate) outliers for the Revenue Growth, the Q-Q Normal plot reveals that they are each just a little over 2 standard deviations away from the mean. It is questionable, therefore, to classify them as true outliers. The results of the 1, 2, and 3 standard deviation ranges, together with the graphs, support a conclusion that the Revenue Growth data could be described with the Normal, in light of the small sample size. Part II: 2. Sample 1 30.13 21.84 13.09 20.46 12.76 25.86 21.34 22.16 16.99 22.00 20.58 11.60 15.14 24.43 15.41 Sample 2 18.43 20.75 9.22 26.43 24.94 16.23 17.88 27.12 23.13 20.06 14.97 22.63 12.27 20.69 22.92 Sample 3 15.16 13.69 17.96 16.73 19.57 21.21 18.65 18.31 22.88 26.95 19.17 21.60 20.34 23.39 26.43 NOTE each student's data should be different due to the nature of random sampling. Sample 1, (n = 15) Histogram of Sample 1 / Data Se t #1 7 6 Standardized Q-Value 5 Frequency 4 3 2 1 0 13.450 17.157 20.864 24.571 28.278 -3 Z-Value Q-Q Normal Plot of Sample 1 / Da ta Set #1 3 Appearance: Not bad, but looks a little skewed to the right 2 1 StatTools Student Version For Academic Use Only 0 -3 -2 -1 -1 -2 0 1 2 3 Sample 2, (n = 15) Histogram of Sample 2 / Data Se t #1 7 6 Standardized Q-Value 5 Frequency 4 3 2 1 0 11.008 14.588 18.168 21.748 25.328 -3 Z-Value Q-Q Normal Plot of Sample 2 / Da ta Set #1 3 2 1 Looks slightly skewed to the left 0 1 2 3 StatTools Student Version For Academic Use Only 0 -3 -2 -1 -1 -2 Sample 3, (n = 15) Histogram of Sample 3 / Data Set #1 6 5 Standardized Q-Value 4 Frequency 3 2 1 0 15.015 17.667 20.319 22.971 25.623 -3 Z-Value Q-Q Normal Plot of Sample 3 / Data Set #1 3 2 1 StatTools Student Version For Academic Use Only 0 -3 -2 -1 -1 -2 0 1 2 3 Actually looks quite good 3. NOTE students do not need to include the actual data for this question. Sample 1, (n = 30) Histogram of Sample 1 / Data Se t #2 8 7 2 6 Standardized Q-Value Frequency 5 4 3 2 1 0 13.152 16.360 19.568 22.776 25.985 29.193 -3 Z-Value 1 Q-Q Normal Plot of Sample 1 / Da ta Set #2 3 Appearance: StatTools Student Version For Academic Use Only 0 -3 -2 -1 -1 -2 0 1 2 3 Not bad; perhaps a little skewed to the right Sample 2, (n = 30) Q-Q Normal Plot of Sample 2 / Data Set #2 Histogram of Sample 2 / Data Se t #2 3 9 8 2 Standardized Q-Value 1 7 6 Frequency 5 4 3 2 1 0 9.400 12.809 16.217 19.625 23.033 26.441 StatTools Student Version For Academic Use Only 0 -3 -2 -1 -1 -2 0 1 2 3 Not bad: perhaps a little skewed to the left -3 Z-Value Sample 3, (n = 30) Histogram of Sample 3 / Data Set #2 12 10 8 Frequency 6 4 2 0 11.217 14.623 18.030 21.436 24.843 28.249 -3 Z-Value Standardized Q-Value Q-Q Normal Plot of Sample 3 / Data Set #2 3 2 1 StatTools Student Version For Academic Use Only 0 -3 -2 -1 -1 -2 0 1 2 3 Quite good 4. NOTE students do not need to include the actual data for this question. Sample 1, (n = 100) Q-Q Normal Plot of Sample 1 / Data Set #3 Histogram of Sample 1 / Data Set #3 3 30 25 Appearance: 2 Standardized Q-Value 1 20 Frequency 15 10 5 0 12.151 14.606 17.061 19.517 21.972 24.428 26.883 29.339 StatTools Student Version For Academic Use Only Very good 0 1 2 3 0 -3 -2 -1 -1 -2 -3 Z-Value Sample 2, (n = 100) Histogram of Sample 2 / Data Set #3 25 Q-Q Normal Plot of Sample 2 / Data Set #3 3 20 Standardized Q-Value 2 1 Exceptional Frequency 15 StatTools Student Version 10 For Academic Use Only 0 -3 -2 -1 -1 -2 0 1 2 3 5 0 10.867 13.665 16.463 19.261 22.059 24.857 27.655 30.454 -3 Z-Value Sample 3, (n = 100) Histogram of Sample 3 / Data Set #3 30 25 20 Frequency 15 10 5 0 10.233 12.826 15.419 18.012 20.605 23.198 25.792 28.385 -3 Z-Value Standardized Q-Value Q-Q Normal Plot of Sample 3 / Data Set #3 3 2 1 Very good 0 1 2 3 StatTools Student Version For Academic Use Only 0 -3 -2 -1 -1 -2 5. We note that the distributions vary in appearance within sample size, and across sample size. By looking at the Q-Q Normal Plots and the Histograms for the samples of size 15, the data does not appear to be particularly Normal (even though we know that we sampled from a Normal distribution) for Sample 1 and Sample 2. By increasing the sample size to 30, the histograms more closely resemble bell shapes and the Q-Q Normal Plots more closely resemble a 45-degree line, but still aren't perfect. When looking at the graphs from the samples of size 100, we can see that the data looks extremely Normal, that is bell shapes and 45-degrees lines are very clear. From this, we can draw the conclusion that, even though all of the samples were taken from a Normal distribution, the larger the sample size, the more closely the data will resemble the parent population. ...
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This note was uploaded on 05/07/2008 for the course BUSMGT 330m taught by Professor Kriska during the Spring '08 term at Ohio State.

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