Regression Analysis

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1. Chi-square Goodness-of-fit Tests Jake is trying to invest his money in stock market, is not sure that he could earn a profit or lose his money when he invests to an AT&T company’s stock or a stock market index, Dow Jones Industry Average. So he called his friend who works at financial consulting company and heard that the monthly positive and negative investment returns on AT&T and Dow Jones Industry Average were historically almost the same. However the economic situation recently has been getting better than in previous years. So before investing he is going to collect related data and prove the hearsay is rooted in fact or not.

a) Chi-square Goodness-of-fit Tests on AT&T In order to test the validity of the hearsay, he collects 54 observations of monthly investment returns on AT&T and DJIA (Dow Jones Industrial Average) from March 2008 to September 2012. After gathering the data, he is going to test whether the number of months that has positive investment returns on AT&T are equal to months of negative returns. Therefore he sets a hypothesis; the null hypothesis H0 :ppositive = pnegative = 0.5 versus the alternative hypothesis Ha:

not H0. The number of months that has the positive returns on AT&T is 32, and the number of months that has the negative returns is 22. In order to test the hypothesis, he should do chi-square test. First of all he calculates the expected value that has the positive and negative returns respectively, that is 54 (n) × 0.5 = 27 in each case as below (The expected value is more than 5, so he can use chi-square test). AT&T (Actual) Positive 32 Negative 22 AT&T (Expected) Positive 27 Negative 27

And he calculates the chi-square value, χ2 = [(32 – 27)2/27] + [(22 – 27)2/27] = 25/27 + 25/27 = 1.8519

Cohort 2- Team 5

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Because χ2 is less than χ20.1 = 2.70544 (degree of freedom = 2 (np) – 1 = 1, np is number of probability, ppositive and pnegative), so he doesn’t reject H0 at 10 % significance level. The p value of Excel calculation is 0.1736 (p value >α = 0.1, so do not reject H0). That means that the number of months of positive return and of negative return is the same.

But he is still suspicious of this result because the actual value of the positive return is bigger than that of the negative return. It looks like nearly twice as many positive returns as negative returns. Therefore he sets an additional hypothesis; the null hypothesis H0 : ppositive = 2 × pnegative versus the alternative hypothesis Ha : ppositive ≠ 2 × pnegative. Then he acquires the expected positive return, 54 (n) × 2 / 3 = 36 and the expected negative return, 54 (n) × 1 / 3 = 18 (The expected value is more than 5, so he can use chi-square test). AT&T (Actual) Positive 32 Negative 22 AT&T (Expected) Positive 36 Negative 18

And he calculates the chi-square value, χ2 = [(32 – 36)2/36] + [(22 – 18)2/18] = 16/36 + 16/18 = 1.3333 Because χ2 is less than χ20.1 = 2.70544 (degree of freedom = 2 (np) – 1 = 1), so he doesn’t reject H0 at 10 % significance level. The p value of Excel calculation is 0.2482 (p value >α = 0.1, so do not reject H0). Therefore the number of months that has the positive return is twice the number of months of negative return.

b) Chi-square Goodness-of-fit Tests on DJIA Jake also would like to test the same hypothesis to DJIA. That means he is going to test whether the number of months that has positive investment returns on DJIA is equal to the months of negative ones. So he sets a hypothesis; the null hypothesis H0 : ppositive = pnegative = 0.5 versus the alternative hypothesis Ha : not H0. The number of months that has the positive returns on DJIA is 34, and the number of months that has the negative returns is 20. In order to test the hypothesis, he should do chi-square test. First of all he calculates the expected value that has the positive and Cohort 2- Team 5 Page 2

negative returns respectively, that is 54 (n) × 0.5 = 27 in each case as below. (The expected value is...
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