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e Chegg Study! Guided Solution × Aplia: Student Question + iiki courses aplia.com at servlet quiz?quizactionatakeQuiz&quiz; probGuid-ONAPC0 Search 8. Logistic regression AaAa Credit card fraud is fraud perpetrated through stolen credit cards or credit card information. For years, credit card issuers have been using data mining and statistical tools to detect fraud. Citibank reported that knowing the type of product or service bought, frequency of purchases, and size and location of transaction can significantly reduce fraud. (Source: Jesus Mena, Investigative Data Mining for Security and Criminal Detection, Butterworth-Heinemann, pp. 250-251) A data-mining analyst at a major credit card company would like to construct and test a simple logistic regression model for detecting credit card fraud using data on card transactions classified as either fraudulent or non-fraudulent. The dependent variable for the model is: y 1 if the transaction is due to credit card fraud; 0 if the transaction is not due to credit card fraud = The independent variables for the model are chosen from the following: X1 = dollar amount of the transaction X2 = number of transactions in the preceding 12 hours x31 if the Standard Industry Code (SIC) for the product or service bought never appeared in the card owner's transaction history: 0 if otherwise x41 if the ZIP code of the transaction never appeared in the card owner's transaction history; 0 if otherwise The analyst would like to test a logistic regression model that predicts credit card fraud using the dollar amount of the transaction, the number of transactions in the preceding 12 hours, and the indicator variable for whether the ZIP code of the transaction never appeared in the card owner's transaction history. (Note: Actual fraud-detection models used by credit card companies are much more complicated than the above, including up to hundreds of independent variables.) The logistic regression equation for the above model is: O E(y)-exp@o + 3x3 + 4x4) Session 56:17 Timeout 6:14 PM 11/28/2017

Explanation / Answer

The Logistics Regression Equation of the above model is:

E(y) = exp(0 + 1 x1 + 2 x2 + 4 x4) / [1+ exp(0 + 1 x1 + 2 x2 + 4 x4)]

The z statistic for the test of the significance of the independent variable x4 is 2.60 ( = 0.1645/0.0632) and the pvalue is (.9953). Since Z > 2 at alpha = 0.05 , we conclude that x4 is significant at significance level = .05

The estimated logit for the regression model is:

G(x1,x2,x4) = exp(0.0297 + 0.0045 x1 + 0.2701 x2 + 0.1645 x4) / [1+ exp(0.0297 + 0.0045 x1 + 0.2701 x2 + 0.1645 x4)]

The odds ratio related to the coefficient 2 of the variable x2 is given by exp(2 ). The estimated value of this odds ratio is 1.31.

Logit (p1) = exp(0.0297 + 0.0045 * 378 + 0.2701 * 5 + 0.1645 * 1) = 25.679

Probaility ( P) = p1/(1+p1) = 0.9625

Logit (p2) = exp(0.0297 + 0.0045 * 378 + 0.2701 * 6 + 0.1645 * 1) = 33.643

Probaility ( P) = p2/(1+p2) = 0.9711

Suppose the dollar amount is 378$, ZIP Code never occurred . When the number of preceding transactions is 5, estimated odds that the transaction is fraudulent is 25.679, and the corresponding estimated probability is 0.9625.

If the number to preceding transactions increases to 6 then, estimated odds is 33.643, and the estimated probability is 0.9711.

The ratio of second odds to first is 1.31.

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