Ka-fu Wong © 2003 1

Dr. Ka-fu Wong

ECON1003Analysis of Economic Data

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GOALS

1. Illustrate the concepts of hypothesis testing.

Additional materialsCard demonstration of hypothesis Card demonstration of hypothesis teststests

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 = 0.5Standard deviation = (0.5*0.5)1/2 = 0.5

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Card experiment results (10 trials)

Trial Card color (B/R) Proportion

1 B 0

2 B 0

3 B 0

4 B 0

5 B 0

6 B 0

7 B 0

8 B 0

9 B 0

10 B 0

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Hypothesis

Hypothesis: =0.5 Alternative Hypothesis: <

0.5

Experimental results: (Number of red cards in

10 trials) / 10 = x P(X=x) =nCxpx(1-p)n-x

= 10Cx(0.5)x(0.5)10-x

　 　 cumulative

X p(X) probability

0 0.00098 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

Is it still possible for the deck of cards to be a standard deck of cards?Not very probable.

Reject the original hypothesis

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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 = 0.5x

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

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.

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What is a Hypothesis?

A Hypothesis is a statement about the value of a population parameter developed for the purpose of testing.

Null Hypothesis H0: 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 H1: 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.

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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”.

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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.

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

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.

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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) =nCxpx(1-p)n-x

= 10Cx(0.5)x(0.5)10-x

　 　 cumulative

X p(X) probability

0 0.00098 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

P-value is the probability of getting what we get.

P-value = 0.00098 in our experiment.

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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, H0 is rejected.

If the p-Value is larger than the significance level, H0 is not rejected.

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Additional materialsCard demonstration of hypothesis Card demonstration of hypothesis teststests