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# Class Note for ECON 2370 at UH

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Date Created: 02/06/15
2 Hypotheses re ect past experience with similar questions educated propositions about cause 3 Multiple hypotheses should be proposed whenever possible One should think of alternative causes that could explain the observation the correct one may not even be one that was thought of 4 Hypotheses should be testable Hypotheses can be proven wrong incorrect but can never be proven or con rmed with absolute certainty Someone in the future with more knowledge may nd a case where the hypothesis is not true U Statistical method 1 U Observe the economy raise a question or set of questions Prepare an answer in the form of a hypothesis HO also known as the null hypothesis Prepare counter responses Ha alternative if the null is proven wrong or incorrect Collect data Specify a statistical test 2 6 Determining the critical regions to reject 7 Obtain the ndings prepare the results My modest example using Spam and eggs Implications from theory what the theory predicts with the respect the differences in the proportion of income spent on a good Income spent on good i 7 Total income Let s look at the value of 7 spent by high income 7mg and the the value of 7 spent by low income 7L0w wow 7111 lt O gt Luxury good now 7mm O gt Normal good wow 7H2in gt O gt Inferior good Test 1 is Spam a luxury good 1 Hypothesis to reject gt 7L0 7mg gt O I wish to reject the notion that the good could be either a normal or inferior good Logically rejecting this hypothesis implies that I fail to reject that it is a luxury good Failing to reject Accepting an outcome Accepting an outcome implies that you have accepted the theory Full bank of tests for Is Spam a luxury good 1a Null Hypothesis to reject gt 7L0 7mg 2 0 1b Alternative hypothesis gt 7L0 7mg lt 0 2a Null Hypothesis to reject gt 7L0 7mg 0 2b Alternative hypothesis gt 7L0 7mg 0 3a Null Hypothesis to reject gt 710 7mm lt 0 3b Alternative hypothesis gt 7L0 7mg gt O The researcher will want to reject Tests 1 and 2 and fail to reject 3 Further comment regarding the three tests Two are one tailed tests one is a two tailed test Speci cs Making Assumptions 1 Type variable categorical interval 2 Type of population binomial nornial nornial given the Central Liniit Theoreni Type of analysis univariate bivariate 00 Large or small sample Differences in variances has an impact on differences in means test using a small sample U 6 Null hypothesis HO and alternative hypothesis one or two tailed test 4 Speci cs Sampling Distribution 1 Standard normal 2 scores 2 Student s t distribution 3 X2 distribution 4 F distribution Speci cs One tailed or two tailed If the hypothesis is an inequality eg u gt 0 u lt 1 we can use a one tail test If we are testing if u is a speci c value the alternative hypothesis is that u is not this value and can be any value in the distribution For this case we use a two tail test Speci cs Choosing a critical region Describes rejection area Answers the questions what are we willing to risk in being wrong Three scenariostwo tailed test Scenario 1 04 20 052 2 10 or p value 010 means that when we reject the hypothesis we reject it with a con dence level of 80 Scenario 2 04 10 052 2 5 or p value 005 means that when we reject the hypothesis we reject it with a con dence level of 90 O1 Scenario 3 04 2 052 2 1 or p value 001 means that when we reject the hypothesis we reject it with a con dence level of 98 Notion of Signi cance table on page 346 Do researchers only report results that are signi cant Risk There are two types 1 Type I error rejecting a hypothesis when in fact it is true 2 Type II error failing to reject a hypothesis when one should reject it Probability of making a Type I error Signi cance level p value tells you the probability of making a Type I error Amount of risk for the three scenarios Highest risk of making a Type I error lowest con dence level taken Lowest risk of making a Type I error highest con dence level taken Probability of making a Type II error The probability of a Type II error is 5 6 Power of your statistical test is given as 1 5 Computing 5 and 16 Suppose your hypothesis test is that HOS 10 A You want to compute a power test to determine the probability of rejecting HO when the alternative mean M C 1 Compute the two con dence interval values The book uses the margin of error values but the example 98 assumes that the signi cance level is 5 My instructions apply for all signi cance levels These values are the endpoints of the Type II region The formula for the con dence interval is 10 l Note function uses m From this you have the left boundary value LBV and right boundary value REV 2 Draw the two graphs The left and right boundary points are points around no Determine where M is relative to these boundaries and determine the rejection area of the new distribution overlapping the acceptance region of the old distribution 7 3 Compute z scores for the two values using the following functions LBV a Zleft boundary S E B Zright boundary Note this set of functions uses Ma 4 Given the drawing above determine the paccepting ha when u M The power of the test or the probability of correctly rejecting HO given that u is M is 15 Relationships between Type I 85 II probabilities and power test 1 Increasing the significance level reduces the confidence interval thus increasing the probability of a Type II error and reducing the power of the test 2 Increasing the sample size decreases the standard error This decreases the probability of a Type II error and increases the power of the test 3 If M is very close to no it weakens the power test Large Sample tests of hypothesis Main points in this chapter 1 Standard method to test research questions 2 Discussion risks involved when decision based on the test is incorrect 3 Detailed discussion of the standard method 4 Application of standard method for research questions using large samples Recall chem lab or your biology lab in high school and the Scienti c Method Observations A good scientist is observant and notices thing in the world around him herself She sees hears or in some other way notices whats going on in the world becomes curious about whats happening and raises a question about it Hypothesis This is a tentative answer to the question an explanation for what was observed The scientist tries to explain what caused what was observed hypo under beneath thesis an arranging Testing 1 Hypotheses are possible causes An hypothesis is not an observation rather a tentative explanation for the observation 1

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