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Published byKelvin Blackmer Modified over 2 years ago

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Topics Today: Case I: t-test single mean: Does a particular sample belong to a hypothesized population? Thursday: Case II: t-test independent means: Are two sample means drawn from an identical population or different populations?

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**Z-Test vs t-Test Z-Test T-Test**

Where population standard deviation and standard error are known Where sample size is > 30 Where the normal curve is the model for the sampling distribution for determining the probability of obtaining our result under the null hypothesis T-Test Where population standard deviation and standard error are not known and have to be estimated from the sample data Where the t distribution is model for sampling distribution for determining probability of our results under the null hypothesis

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Z and t Compared

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**Z and t Sampling Distributions**

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**Degrees of Freedom (dfs)**

t distributions differ according to their degrees of freedom (based on the sample size) For single sample case the t distribution is based on n-1 dfs For 2 sample case the t distribution for differences is based on N-2 dfs (i.e., n -1 for each sample)

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**Hypothesis Testing Using t-Tests**

Identify population of interest Draw samples (preferably probability samples) Set up null and alternative hypotheses Select level of significance (e.g., .05, .01) Calculate sample statistic (e.g. mean) Calculate standard error of the sample statistic Convert observed mean to standard error points Determine t-critical value (based on level of significance chosen) Compare t-observed value against the t-critical value Decide: “Reject” or “Do not reject” null hypothesis

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**Sampling Distribution of Means: Standard Errors, Critical Values, and Ps**

t Distribution Sample Size =31 df=30 = .05 or .01 Two tailed Test -2se -1se +1se +2se Critical Values -2.75se -2.04se +2.04se +2.75se p = < = .05 = outside 2.04 on either end < = .01 = outside of 2.57 on either end

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**Sampling Distribution of Means: Standard Errors, Critical Values, and Ps**

t Distribution Sample Size =31 df=30 = .05 or .01 One tailed test u -2se -1se +1se +2se Critical Values se +2.457se p = < = .05 < = .01

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**Sampling Distribution of Means: Standard Errors, Critical Values, and Ps**

t Distribution Sample Size =31 df=30 = .05 or .01 One tailed test u -2se -1se +1se +2se Critical Values -2.457se se p = < = .01 < = .05

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Case I: t-Test Does a particular sample belong to a hypothesized population? Draw single sample from population Calculate sample statistic such as mean Test null hypothesis of no difference between sample and population mean

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**Case I: Assumptions Scores randomly sampled from some population**

Scores in the population are normally distributed

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**Example t-Test Single Sample: SAT Data: UCLA from OAS**

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**t-Test for Single Sample Designs**

Set Null Hypothesis: Set Alternative Hypothesis: Decide Significance Level: .01 Compute Standard Error: Compute tobserved Locate tcritical with N-1df: Decide Reject H0 if tobserved >= tcritical : Do Not Reject H0 if tobserved < tcritical : Conclude:

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**UCLA: Sampling Distribution Picture**

t Distribution Sample Size =10 df=9 = .01 One tailed test 3.95 -2se -1se +1se +2se = 1000 Critical Values +2.82se p = = < .01

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**X - tcritical(.01/1,9) (sx) <= <= X+ tcritical(.01/1,9) (sx)**

99% Confidence Interval X - tcritical(.01/1,9) (sx) <= <= X+ tcritical(.01/1,9) (sx) (21.26) <= <= (21.26) <= <= <= <= Can feel 99% confident that this interval includes the population mean for the UCLA undergraduates. Since it does not include the “known” population mean of 1000 we can conclude that UCLA undergraduates have on average higher total SAT scores than the population of high school graduates taking the SAT.

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1 Level of Significance α is a predetermined value by convention usually 0.05 α = 0.05 corresponds to the 95% confidence level We are accepting the risk.

1 Level of Significance α is a predetermined value by convention usually 0.05 α = 0.05 corresponds to the 95% confidence level We are accepting the risk.

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