Dr Kafu Wong ECON 1003 Analysis of Economic

Additional materials Card demonstration of hypothesis tests GOALS 1. Illustrate the concepts of hypothesis

Card experiment n We are going to perform an experiment on a deck of

Card experiment results (10 trials) Ka-fu Wong © 2003 Trial Card color (B/R) Proportion

Hypothesis n Hypothesis: =0. 5 n Alternative Hypothesis: < 0. 5 n Experimental results:

Hypothesis X p(X) 0 0. 50000 1 0. 25000 n How many draws did

What is a Hypothesis? n A Hypothesis is a statement about the value of

What is the level of significance? n Sometimes we may want to set the

What is a critical value? n corresponds to a “critical value”. n Critical value:

What is p-value? n Hypothesis: =0. 5 n Alternative Hypothesis: < 0. 5 n

p-Value in Hypothesis Testing n A p-Value is the probability, assuming that the null

Additional materials Card demonstration of hypothesis tests - END - Ka-fu Wong © 2003

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Card experiment n We are going to perform an experiment on a deck of 52 cards. n Count the actual number of red cards out of 10 trials (with replacement). n What is the probability of getting a red card on any trial? n Hypothesis: p=0. 5 n. Expected value = 0. 5 n. Standard deviation = (0. 5*0. 5)1/2 = 0. 5 Ka-fu Wong © 2003 3

Hypothesis n Hypothesis: =0. 5 n Alternative Hypothesis: < 0. 5 n Experimental results: n (Number of red cards in 10 trials) / 10 = x n P(X=x) =n. Cxpx(1 -p)n-x = 10 Cx(0. 5)10 -x Is it still possible for the deck of cards to be a standard deck of cards? Not very probable. Reject the original hypothesis Ka-fu Wong © 2003 cumulative X p(X) probability 0 0. 00098 1 0. 00977 0. 01074 2 0. 04395 0. 05469 3 0. 11719 0. 17188 4 0. 20508 0. 37695 5 0. 24609 0. 62305 6 0. 20508 0. 82813 7 0. 11719 0. 94531 8 0. 04395 0. 98926 9 0. 00977 0. 99902 10 0. 00098 1. 00000 5

Hypothesis X p(X) 0 0. 50000 1 0. 25000 n How many draws did it take before the class started feeling uncomfortable with the outcome? n The probability that we do not get any red in a sequence of x trials is P(black)x = 0. 5 x 2 0. 12500 3 0. 06250 4 0. 03125 5 0. 01563 6 0. 20508 7 0. 00781 Most of us were ready to reject the deck as fair after 4 to 5 draws. 8 0. 00391 9 0. 00195 10 0. 00098 n Hypothesis: =0. 5 n Alternative Hypothesis: < 0. 5 We had a good feel of how improbable the hypothesis was. Ka-fu Wong © 2003 6

What is a Hypothesis? n A Hypothesis is a statement about the value of a population parameter developed for the purpose of testing. n Null Hypothesis H 0: A statement about the value of a population parameter. n The probability of getting red card on any trial is 0. 5. n The proportion of red cards in the deck is 0. 5. n Alternative Hypothesis H 1: A statement that is accepted if the sample data provide evidence that the null hypothesis is false. n The probability of getting red card on any trial is less than 0. 5. n The probability of getting red card on any trial is not 0. 5. Ka-fu Wong © 2003 7

What is the level of significance? n Sometimes we may want to set the limits of what we will accept ahead of time. lets us set the limit of where we feel something will be improbable. n Level of Significance ( ): The probability of rejecting the null hypothesis when it is actually true. n If, under the null hypothesis, the probability of observing the sample is less than , the null is rejected. n A pre-set corresponds to a “critical value”. Ka-fu Wong © 2003 8

What is a critical value? n corresponds to a “critical value”. n Critical value: The dividing point between the region where the null hypothesis is rejected and the region where it is not rejected. How many draws did it take before the class started feeling uncomfortable with the outcome? Most of us were ready to reject the deck as fair after 4 to 5 draws. If we were ready to reject the deck as fair after 4 draws, the critical value is 4. The level of significance is about 0. 03125. Ka-fu Wong © 2003 X p(X) 0 0. 50000 1 0. 25000 2 0. 12500 3 0. 06250 4 0. 03125 5 0. 01563 6 0. 20508 7 0. 00781 8 0. 00391 9 0. 00195 10 0. 00098 9

What is p-value? n Hypothesis: =0. 5 n Alternative Hypothesis: < 0. 5 n Experimental results: n (Number of red cards in 10 trials) / 10 = x n P(X=x) =n. Cxpx(1 -p)n-x = 10 Cx(0. 5)10 -x P-value is the probability of getting what we get. P-value = 0. 00098 in our experiment. Ka-fu Wong © 2003 cumulative X p(X) probability 0 0. 00098 1 0. 00977 0. 01074 2 0. 04395 0. 05469 3 0. 11719 0. 17188 4 0. 20508 0. 37695 5 0. 24609 0. 62305 6 0. 20508 0. 82813 7 0. 11719 0. 94531 8 0. 04395 0. 98926 9 0. 00977 0. 99902 10 0. 00098 1. 00000 10

p-Value in Hypothesis Testing n A p-Value is the probability, assuming that the null hypothesis is true, of finding a value of the test statistic at least as extreme as the computed value for the test. n If the p-Value is smaller than the significance level, H 0 is rejected. n If the p-Value is larger than the significance level, H 0 is not rejected. Ka-fu Wong © 2003 11