1. Ka-fu Wong © 2003 1 Dr. Ka-fu Wong ECON1003 Analysis of Economic Data

2. Ka-fu Wong © 2003 2 l GOALS 1.Illustrate the concepts of hypothesis testing. Additional materials Card demonstration of hypothesis tests

3. Ka-fu Wong © 2003 3 Card experiment We are going to perform an experiment on a deck of 52 cards. Count the actual number of red cards out of 10 trials (with replacement). What is the probability of getting a red card on any trial? Hypothesis: p=0.5 Expected value = 0.5 Standard deviation = (0.5*0.5) 1/2 = 0.5

4. Ka-fu Wong © 2003 4 Card experiment results (10 trials) TrialCard color (B/R)Proportion 1B0 2B0 3B0 4B0 5B0 6B0 7B0 8B0 9B0 10B0

5. Ka-fu Wong © 2003 5 Hypothesis Hypothesis: =0.5 Alternative Hypothesis: < 0.5 Experimental results: (Number of red cards in 10 trials) / 10 = x P(X=x) = n C x p x (1-p) n-x = 10 C x (0.5) x (0.5) 10-x cumulative Xp(X)probability 00.00098 10.009770.01074 20.043950.05469 30.117190.17188 40.205080.37695 50.246090.62305 60.205080.82813 70.117190.94531 80.043950.98926 90.009770.99902 100.000981.00000 Is it still possible for the deck of cards to be a standard deck of cards? Not very probable. Reject the original hypothesis

6. Ka-fu Wong © 2003 6 Hypothesis Hypothesis: =0.5 Alternative Hypothesis: < 0.5 How many draws did it take before the class started feeling uncomfortable with the outcome? The probability that we do not get any red in a sequence of x trials is P(black) x = 0.5 x Xp(X) 00.50000 10.25000 20.12500 30.06250 40.03125 50.01563 60.20508 70.00781 80.00391 90.00195 100.00098 Most of us were ready to reject the deck as fair after 4 to 5 draws. We had a good feel of how improbable the hypothesis was.

7. Ka-fu Wong © 2003 7 What is a Hypothesis? A Hypothesis is a statement about the value of a population parameter developed for the purpose of testing. Null Hypothesis H 0 : A statement about the value of a population parameter. The probability of getting red card on any trial is 0.5. The proportion of red cards in the deck is 0.5. Alternative Hypothesis H 1 : A statement that is accepted if the sample data provide evidence that the null hypothesis is false. The probability of getting red card on any trial is less than 0.5. The probability of getting red card on any trial is not 0.5.

8. Ka-fu Wong © 2003 8 What is the level of significance? 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. Level of Significance (  ): The probability of rejecting the null hypothesis when it is actually true. If, under the null hypothesis, the probability of observing the sample is less than , the null is rejected. A pre-set  corresponds to a “critical value”.

9. Ka-fu Wong © 2003 9 What is a critical value?  corresponds to a “critical value”. Critical value: The dividing point between the region where the null hypothesis is rejected and the region where it is not rejected. Xp(X) 00.50000 10.25000 20.12500 30.06250 40.03125 50.01563 60.20508 70.00781 80.00391 90.00195 100.00098 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.

10. Ka-fu Wong © 2003 10 What is p-value? Hypothesis: =0.5 Alternative Hypothesis: < 0.5 Experimental results: (Number of red cards in 10 trials) / 10 = x P(X=x) = n C x p x (1-p) n-x = 10 C x (0.5) x (0.5) 10-x cumulative Xp(X)probability 00.00098 10.009770.01074 20.043950.05469 30.117190.17188 40.205080.37695 50.246090.62305 60.205080.82813 70.117190.94531 80.043950.98926 90.009770.99902 100.000981.00000 P-value is the probability of getting what we get. P-value = 0.00098 in our experiment.

11. Ka-fu Wong © 2003 11 p-Value in Hypothesis Testing 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. If the p-Value is smaller than the significance level, H 0 is rejected. If the p-Value is larger than the significance level, H 0 is not rejected.

12. Ka-fu Wong © 2003 12 - END - Additional materials Card demonstration of hypothesis tests