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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.

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Presentation on theme: "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."— Presentation transcript:

1 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?

2 Z-Test vs t-Test Z-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

3 Z and t Compared

4 Z and t Sampling Distributions

5 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)

6 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 Hypothesis Testing Using t-Tests

7 Sampling Distribution of Means: Standard Errors, Critical Values, and Ps  +2se -2se+1se-1se +2.04se+2.75se-2.75se-2.04se < =.01 = outside of 2.57 on either end< =.05 = outside 2.04 on either end p = Critical Values t Distribution Sample Size =31 df=30  =.05 or.01 Two tailed Test

8 Sampling Distribution of Means: Standard Errors, Critical Values, and Ps u +2se-2se+1se-1se se+2.457se < =.01< =.05 p = Critical Values t Distribution Sample Size =31 df=30  =.05 or.01 One tailed test

9 Sampling Distribution of Means: Standard Errors, Critical Values, and Ps u +2se -2se+1se-1se se-2.457se < =.01< =.05 p = Critical Values t Distribution Sample Size =31 df=30  =.05 or.01 One tailed test

10 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

11 Case I: Assumptions Scores randomly sampled from some population Scores in the population are normally distributed

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

13 t-Test for Single Sample Designs Set Null Hypothesis: Set Alternative Hypothesis: Decide Significance Level:.01 Compute Standard Error: Compute t observed Locate t critical with N-1df: Decide –Reject H 0 if t observed >= t critical : –Do Not Reject H 0 if t observed < t critical : Conclude:

14 UCLA: Sampling Distribution Picture  +2se-2se+1se-1se +2.82se = <.01 p = Critical Values t Distribution Sample Size =10 df=9  =.01 One tailed test 3.95 = 1000

15 99% Confidence Interval X - t critical(.01/1,9) (s x ) <=  <= X+ t critical(.01/1,9) (s x ) (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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