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Name:BUAD 300Second Mini-ProjectPROBLEM STATEMENTThis report presents an analysis of whether for this casino, giving patrons free-play on slot-machines motivates them to play their own funds after the free-play is exhausted. Given that the average hold for this casino is 9%, the coin-in per $1 of free-play redeemed must exceed $11.11 if players gamble with their own funds after their free-play is exhausted. (See the appendix to this report for an explanation of this $11.11 figure.) To explore this return to free-play, day-of-the-week effects, seasonality, and holidays are measured and controlled for, and the returns to free-play are measured independent of these controlled-for effects. In addition, a forecast for the next day’s coin-in is presented. This forecast assumes $20,000 of free play redeemed. DATA ANALYZEDThe data analyzed included XXX consecutive daily observations on coin-in and redeemed free play. This data set extends from the initial data of XXXX to the last date of XXXX. Table 1 presents the means and standard deviations of coin-in and free play, as well the minimum and maximums of each. Table 2 presents the day-of-the-week means for coin-in and free play. The correlation coefficient between coin-in and free play is XXX.Table 1:Basic Statistics for Coin-In and Promo-PlayFor xxxxxx Through xxxxxxMeanMaximumMinimumStandard DeviationCoin-InFree PlayTable 2:Mean Statistics Per Day-of-the Week for Coin-In and Promo-PlayFor xxxxxx Through xxxxxxSundayMondayTuesdayWednesdayThursdayFridaySaturdayCoin-InFree PlaySTATISTICAL METHODSOLS regression is used to estimate model equations (1) and (2) where ∈ is the random-error term with the usual OLS assumptions applying. The regression coefficients m, u, w, r, f, s are for the respective day-of-the-week dummy variables (0 or 1), where the coefficient for Sunday is not included in order to avoid the dummy variable trap. N is the regression coefficient for the seasonal dummy (1 for days December 26 through March 31, and 0 for other days), and H is the regression coefficient for the holiday dummy (1 for December 31 and January 1, and 0 for other days). An F-statistic is used for testing hypothesis (i) given below. An F-statistic is also used to test hypothesis (ii) given below, i.e., whether model (2) has greater explanatory power than model (1). A t-statistic is also utilized for testing the magnitude of the coefficient β1, as presented in hypothesis (iii) given below. To test whether the t-test is appropriate, a χ2 test for normality of the residual errors associated with the OLS estimate of model equation (2) is presented as hypothesis test (iv). Hypothesis (v) is tested via a nonparametric runs analysis.Equation (3) gives the predicted value for coin-in given a prediction for promo, and (4) gives the confidence interval for that prediction.Coin-int = β0 + m(0 or 1) + u(0 or 1) + w(0 or 1) + r(0 or 1) + f((0 or 1) + s(0 or 1) + + N(0 or 1) + H(0 or 1) + ∈t (1)Coin-int = β0 + m(0 or 1) + u(0 or 1) + w(0 or 1) + r(0 or 1) + f((0 or 1) + s(0 or 1) + + N(0 or 1) + H(0 or 1) + β1Free-Playt + ∈t (2)E(Coin-in) = β0 + …. + β1 E(Free-Play) (3) 95% confidence interval: E(Coin-in) ± t2.5% (S.E. of Prediction)(4)H0: There is no day-of-the-week, seasonality or holiday effects, i.e. m = u = w = r = f = s = N = H = 0.Tested with an F-statistic, α = 5%.H0: The OLS estimate of model equation (2) has greater explanatory power than (1).Tested with an F-statistic, α = 5%.H0: β1 > 11.11.Tested with a t-statistic, α = 5%.H0: The residual errors associated with the OLS estimate of (2) are normally distributed. Tested with a χ2 statistic, α = 5%.H0: The residual errors are independent of one another. This is tested with a nonparametric runs analysis.Predictions of the next day’s coin-in, with an associated confidence interval, utilize equations (3) and (4) above.STATISTICAL RESULTSThe results of the OLS regression estimate for (1) are presented in Table 3. As shown, the coefficients r, f, s, N and H are significantly different from zero at a significance level of 95%. The coefficients m, u, and w are not significantly different from zero at a significance level of 95%. In addition, given the F-statistic and its associated p-value, hypothesis (i) is rejected at an α of 5%. Note that the p-value associated with this F statistic is the probability that there is no day-of-the-week, seasonal or holiday effects.Table 3OLS Regression Results for Estimation of Equation (1)OLS CoefficientStandard Errort-statisticp-valueβ0muwrfsNHR2 = XXXX,F = XXXX with associated p-value = XXTable 4 presents the OLS estimates of model equation (2). Results of hypothesis test (iii) are presented below. Results of OLS “analysis of variance” for both (1) and (2) are attached.Table 4OLS Regression Results for Estimation of Equation (2)OLS CoefficientStandard Errort-statisticp-valueβ0muwrfsNHβ1R2 = XXXX,F = XXXX with associated p-value = XXH0: The OLS estimate of model equation (2) has greater explanatory power than (1).Fcalculated = SSE for model 1- SSE for model (2)SSE for model (2)d.f. = XX-YYYY/WW = QQFcritical, 5% = XXDecision rule: If Fcalculated > Fcritical then do not reject H0.Decision: XXXXXXThe results of the t-test for hypothesis (iii) are presented below.H0: β1 > 11.11.tcalculated = OLS β1- Hyp. β1Std.Error = XX- 11.11YY = WWtcritical, 95% = YYYYDecision Rule: If tcritical < tcalculated then do not reject H0.Decision: XXXXXXThe test for hypothesis (iv), that the OLS errors for model (2) are normally distributed, is presented below. For this test, the residual errors were standardized, and frequency counts for various ranges are presented in Table 5 under column f. Each range has an expectation of 10% of the total number of observations assuming normality, and as listed under column e. A test of hypothesis (v) is also presented below.Table 5χ2 Frequency Table for Various Ranges for OLS Residual Errors for Estimates of Equation (2)Range for Std. Errorsfef-e(f-e)2(f-e)2/eχ2calculated = ∑(f-e)2/e = XXH0: OLS residual errors are normal.χ2calculated = XXχ2critical = XXDecision Rule: If χ2critical > χ2calculated then do not reject H0.Decision: YYYYYY.H0: The residual errors are independent of one another.Allow:u = total # of runs of positive and negative errors = Xn+ = # of positive errors = Xn- = # of negative errors = XE(u) = 2n+n-n++ n-+ 1 = XVar(u) = 2n+n-2n+n- -n+ - n-(n++ n-)2( n++ n-- 1) = Xz = u-E(u)√Var(u) =X-X√XX = XXDecision rule: If zcalc. ≥ zα/2 then reject H0.Decision: YYYYY. Given the OLS results for model (2), and given that the next day in the data set is a Monday, then given a $20,000 prediction for free play, the predicted coin-in is $XXXX. The 95% confidence interval is $XXXX to $YYYY.Table 6 presents the significant outliers given the estimate of equation (2).Table 6: Dates of Significant Outliers1Dates of Significant Negative ResidualsStandardized ResidualsDates of Significant Positive ResidualsStandardized Residuals1 Significant residuals are those with standardized absolute magnitudes greater than 1.96.CONCLUSION{Write your own non-technical conclusion, similar to the conclusion in Project 1. Be sure to indicate if on average patrons are playing their own funds after free-play is exhausted, and if so present an estimate of the amount they play of their own funds if any.Attach your Figure 1, your Appendix, bibliography, Minitab output, and a printout of your data showing your day-of-the-week, seasonal and holiday dummy variables.}AppendixCasinos often give free-play, called “promo,” to slot machine players. They do this to induce players to come to their facility rather than a competitor’s facility. Casinos usually keep 9% of the amount played in slots. This means that when promo-play is given, it will generate coin-in that is, on average, 11 times the actual promo granted. “Coin-in” is the gross revenue figure for a slot machine. This multiple of 11 occurs because for each run of the machine, only an average of 9% of the amount gambled is kept by the casino, with the rest available for further play. Once the promo play is exhausted, then the coin-in generated by the free play is given implicitly by (1), and explicitly by (2). Promo= .09(coin in)(1)Coin in = (1/.09)Promo = 11.11(Promo)(2)Casinos that give promo play hope that once the free-play is exhausted, then on average customers will play their own money so that $1 of promo will generate more than $11.11 in coin-in.ReferencesAnderson, D. and D. Sweeney, T. Williams, J. Camm, and J. Cochran (2014), Statistics for Business and Economics, 12th edition, South-Western Cengage, Mason, OH.Freund, John E. (1971), Mathematical Statistics, second edition, Prentice-Hall, Englewood Cliffs, NJ. ................
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