19.25 19.25 19.25 19.435 19.435 19.435 19.66 19.66 19.66 20.04 20.04 20.04 20.76 20.76 20.76 19.25 20.76 19.435 20.04 19.435 20.04 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 1 1 0.5 0.5 1.5 1.5 Year 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 Admissions (millions) Y Screens Theatres Films Screened Real Ticket Price Capacity ('000s) 28.9 822 702 223 19.71 377 29.7 742 573 194 19.82 324 35.5 676 509 239 19.9 295 30.8 645 506 259 19.63 303 37.4 712 520 281 19.66 285 39 772 501 291 19.43 286 43 851 510 252 20.41 295 46.9 885 522 240 19.25 301 47.2 906 510 229 19.39 296 55.5 951 518 259 19.44 300 68.1 1028 537 252 20.76 312 69.9 1137 557 253 20.54 332 73.9 1251 551 281 20.21 356 76 1422 563 285 20.08 387 80 1576 567 273 19.84 413 88 1748 580 255 19.54 446 82.2 1817 554 250 19.38 453 92.5 1855 547 245 19.3 463 92.5 1872 547 258 19.52 464 89.8 1907 553 268 19.56 471 91.5 1909 520 318 20 460 Admissions Mean Standard Error Median Mode Standard Deviation Sample Variance Kurtosis Skewness Range Minimum Maximum Sum Count Screens 61.823809524 1213.524 5.1518046064 103.6609 68.1 1028 92.5 #N/A 23.608534575 475.034 557.36290476 225657.3 -1.6663464306 -1.569617 -0.0555289741 0.410014 63.6 1264 28.9 645 92.5 1909 1298.3 25484 21 21 Theatres Films screened Real ticket price 545.0952 9.458406 547 520 43.34386 1878.69 8.375822 2.445313 201 501 702 11447 21 257.380952381 19.7795238095 5.7695738022 0.0919976092 255 19.66 259 #N/A 26.4395086764 0.4215860077 699.047619048 0.1777347619 1.2563262857 0.0850433316 -0.0570838371 0.9160032801 124 1.51 194 19.25 318 20.76 5405 415.37 21 21 Capacity ('000s) 362.8095238095 15.5708242201 332 295 71.3544806215 5091.461904762 -1.5829969189 0.4435730696 186 285 471 7619 21 Frequencies (Admissions (millions)) BINS Frequency Percentage Cumulative % 25 0 0.00% 0.00% 35 3 14.29% 14.29% 45 4 19.05% 33.33% 55 2 9.52% 42.86% 65 1 4.76% 47.62% 75 3 14.29% 61.90% 85 3 14.29% 76.19% 95 5 23.81% 100.00% The distribution is left skewed Median 68.1 Mean 61.82 Histogram 6 5 4 Frequency 3 2 1 0 25 35 45 55 65 BINS 75 35 Histogram 120.00% 100.00% 80.00% 60.00% 40.00% 20.00% 45 55 65 BINS 75 85 95 0.00% Frequency Cumulative % Five-number Summary Minimum 19.25 First Quartile 19.435 Median 19.66 Third Quartile 20.04 Maximum 20.76 Box-and-whisker Plo 2 1 0 19 19.2 19.4 19.6 19.8 Box-and-whisker Plot - Real Ticket Price 19.6 19.8 20 20.2 20.4 20.6 20.8 21 Q) What is the likelihood that the admissions are greater than 70 million if the real price of tickets exceeds $20.00? Are adm A) There is a 40% probability. Admissions are statistically independent of price Contingency tables Year 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 Admissions (millions) Real Ticket Price 28.9 19.71 29.7 19.82 35.5 19.9 30.8 19.63 37.4 19.66 39 19.43 43 20.41 46.9 19.25 47.2 19.39 55.5 19.44 68.1 20.76 69.9 20.54 73.9 20.21 76 20.08 80 19.84 88 19.54 82.2 19.38 92.5 19.3 92.5 19.52 89.8 19.56 91.5 20 P(B) - Price exceeds $20 P(B') - Price equal to or less than $20 Joint and marginal probability P(B) - Price exceeds $20 P(B') - Price equal to or less than $20 Conditional probability P(B) - Price exceeds $20 P(B') - Price equal to or less than $20 1 2 xceeds $20.00? Are admissions statistically independent of price? Use a Contingency Table. P(A) - Admissions greater than 70m P(A')Admissions equal to or less than 70m 2 7 9 P(A) - Admissions greater than 70m 3 9 12 P(A')Admissions equal to or less than 70m 0.0952380952 0.3333333333 0.4285714286 P(A) - Admissions greater than 70m 5 16 21 0.1428571429 0.238095 0.4285714286 0.761905 0.5714285714 1 P(A')Admissions equal to or less than 70m 0.4 0.4375 0.6 0.5625 Conditional probability P(A|B) = P(A and B)/P(B) Independence P(A, given that B occurs) = P(A) P(B, given that A occurs) = P(B) P(A and B) = P(A)*P(B) x x x *EXPLAIN FURTHER Joint Marginal Confidence Interval Estimate for the Mean Data Sample Standard Deviation Sample Mean Sample Size Confidence Level 69.551552255 357.95 20 95% Intermediate Calculations Standard Error of the Mean 15.552199878 Degrees of Freedom 19 t Value 2.0930240544 Interval Half Width 32.551128444 Confidence Interval Interval Lower Limit Interval Upper Limit *Comment on above*!!! 325.40 390.50 Admissions ('000s) Original Taiwan 2008 80 2009 88 2010 82.2 2011 92.5 2012 92.5 2013 89.8 2014 91.5 84 84 84 84 84 84 84 DOUBLE CHECK ALL OF THIS Descriptive statistics Original Mean Standard Error Median Mode Standard Deviation Sample Variance Kurtosis Skewness Range Minimum Maximum Sum Count 88.071 1.912 89.8 92.5 5.06 25.60 -0.88 -0.92 12.5 80 92.5 616.5 7 t Test for Hypothesis of the Mean Null Hypothesis Data m > Level of Significance Sample Size Sample Mean Sample Standard Deviation 84 0.05 7 88.07 5.06 Intermediate Calculations Standard Error of the Mean 1.9125002334 Degrees of Freedom 6 t Test Statistic 2.1281043154 Lower-Tail Test Lower Critical Value -1.9431802805 p-Value 0.9612940976 Do not reject the null hypothesis Calculations Area For one-tailed tests: TDIST valu 0.038706 1-TDIST va 0.961294 Regression Analysis Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.9892502378 0.978616033 0.971488044 3.9864152596 21 ANOVA df Regression Residual Total SS MS F 5 10908.8854959 2181.7770992 137.29202341 15 238.372599327 15.891506622 20 11147.2580952 Coefficients -71.8811655806 0.0878369174 0.045087957 0.0017863693 5.2766367818 -0.2819484599 Intercept Screens Theatres Films Screened Real Ticket Price Capacity ('000s) Standard Error t Stat P-value 47.5443639849 -1.5118756369 0.1513453769 0.0094386683 9.3060710545 1.27616E-007 0.0347085942 1.2990430181 0.2135431732 0.0415134777 0.0430310685 0.9662443602 2.4119007809 2.1877503518 0.0449329861 0.0673168093 -4.1883812211 0.0007912603 RESIDUAL OUTPUT Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Predicted Admissions (millions) Y 30.0788282787 32.7074221456 32.7035792476 26.2008187121 37.9897950842 40.9256277199 50.834435313 46.5279201311 49.9602612997 53.4632553182 64.6526435901 68.3305838075 69.6154298158 75.7573786164 80.8461265097 85.6207748872 87.6824023563 87.4540421309 Residuals -1.1788282787 -3.0074221456 2.7964207524 4.5991812879 -0.5897950842 -1.9256277199 -7.834435313 0.3720798689 -2.7602612997 2.0367446818 3.4473564099 1.5694161925 4.2845701842 0.2426213836 -0.8461265097 2.3792251128 -5.4824023563 5.0459578691 19 20 21 89.8494041601 2.6505958399 91.4495139566 -1.6495139566 95.6497569193 -4.1497569193 Significance F 5.651799E-012 Lower 95% Upper 95% -173.21957859 29.457247429 0.0677188723 0.1079549626 -0.0288916604 0.1190675744 -0.0866975139 0.0902702526 0.1357919585 10.417481605 -0.4254308425 -0.1384660773 *p-value is less than 0.05 so that means we need to reject the null hypothesis? eject the null hypothesis