lesson9 - Confidence Intervals and Tests of Hypothesis (more than 1 sample)

# Lesson9 - Confidence Intervals and Tests of Hypothesis (more than 1 sample)

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Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson9-1 Lesson 9: Confidence Intervals and Tests of  Confidence Intervals and Tests of  Hypothesis Hypothesis Two or more samples Two or more samples

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Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson9-2 The most important part of testing  hypothesis Suppose we are interested in testing whether the population  parameter ( θ ) is equal to k. H 0 θ  = k  H 1 θ    k  First , we need to get a sample estimate (q) of the population  parameter ( θ ). Second , we need to identify the sampling distribution of q,  including its mean and variance. Third , we know in most cases, the test statistics will be in the  following form: t=(q-k)/ σ q σ q  is the standard deviation of q  under the null .  The form  of  σ depends on what q is.  If depends on k   powerful  test  Fourth , given the level of significance, determine the rejection  region. Null hypothesis(certain numbers that can be illustrated and constructed): Equals to something Larger or equal to something Random variable K is the assumed value of mean Only an example of standardization, depends on the type of population parameter
Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson9-3 Testing a two-sided hypothesis at 5% level of  significance θ 0 q z=(q-  θ 0 )/std(q) is approximately normally distribution under CLT.  α /2 1.96 Rejection region -1.96 α /2 Rejection region θ 0 +1.96* σ q θ -1.96* σ q Assumed value of null hypothesis

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Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson9-4 The most important part of constructing  confidence intervals Suppose we are interested in constructing a (1- α )*100%  confidence interval about the unknown the population  parameter ( θ ), based on some sampling information. First , we must have a sample estimate (q) of the population  parameter ( θ ). Second , we need to identify the sampling distribution of q,  including its mean and variance. Third , we know in most cases, the following statistics will be  approximately normal or student-t distributed: t=(q-k)/ σ q σ q  is the standard deviation of q  under the null .  The form  of  σ depends on what q is.  Fourth , given the confidence level, determine the upper and  lower confidence limit for  θ . q ± t α /2 * σ q                ???
Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson9-5 Constructing a 95% confidence interval for  θ q * q z=(q-  θ )/std(q) is approximately normally distribution under CLT.  α /2 1.96 Upper limit -1.96 α /2 lower limit q * +1.96* σ q q * -1.96* σ q q*: estimate of θ from a sample.

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## Lesson9 - Confidence Intervals and Tests of Hypothesis (more than 1 sample)

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