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Understanding Significance Testing and Confidence Intervals in Statistics

This lecture focuses on significance testing, particularly for proportions and means, comparing between two populations. We explore hypotheses setup, significance levels, the role of sample size, and the process of calculating the test statistics and p-values. The relationship between confidence intervals and hypothesis testing is emphasized, as well as methods for handling small sample sizes and unequal variances. Real-world application is illustrated through examples, including data from the General Social Survey, to analyze differences in population means over time.

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Understanding Significance Testing and Confidence Intervals in Statistics

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  1. STA 291Summer 2010 Lecture 20 Dustin Lueker

  2. Example • If the null hypothesis is rejected at a 2% level of significance, will the null be rejected at a 5% level of significance? • Yes • No • Maybe STA 291 Summer 2010 Lecture 20

  3. Example • If the null hypothesis is rejected at a 2% level of significance, will the null be rejected at a 1% level of significance? • Yes • No • Maybe STA 291 Summer 2010 Lecture 20

  4. Significance Test for a Proportion • Same process as with population mean • Value we are testing against is called p0 • Test statistic • P-value • Calculation is exactly the same as for the test for a mean • Sample size restrictions: STA 291 Summer 2010 Lecture 20

  5. Testing Difference Between Two Population Proportions • Similar to testing one proportion • Hypotheses are set up like two sample mean test • H0:p1-p2=0 • Same as H0: p1=p2 • Test Statistic STA 291 Summer 2010 Lecture 20

  6. Testing the Difference Between Means from Different Populations • Hypothesis involves 2 parameters from 2 populations • Test statistic is different • Involves 2 large samples (both samples at least 30) • One from each population • H0: μ1-μ2=0 • Same as H0: μ1=μ2 • Test statistic STA 291 Summer 2010 Lecture 20

  7. Example • In the 1982 General Social Survey, 350 subjects reported the time spent every day watching television. The sample yielded a mean of 4.1 and a standard deviation of 3.3. • In the 1994 survey, 1965 subjects yielded a sample mean of 2.8 hours with a standard deviation of 2. • Set up hypotheses of a significance test to analyze whether the population means differ in 1982 and 1994 and test at α=.05 using the p-value method. STA 291 Summer 2010 Lecture 20

  8. Correspondence Between Confidence Intervals and Tests • Constructing a confidence interval to do a hypothesis test with 2 samples works the same as it did when we were dealing with 1 sample • The confidence interval shows plausible values for the difference between the two means STA 291 Summer 2010 Lecture 20

  9. Small Sample Tests for Two Means • Used when comparing means of two samples where at least one of them is less than 30 • Normal population distribution is assumed for both samples • Equal Variances • Both groups have the same variability • Unequal Variances • Both groups may not have the same variability STA 291 Summer 2010 Lecture 20

  10. Small Sample Test for Two Means, Equal Variances • Test Statistic • Degrees of freedom • n1+n2-2 STA 291 Summer 2010 Lecture 20

  11. Small Sample Confidence Interval for Two Means, Equal Variances • Degrees of freedom • n1+n2-2 STA 291 Summer 2010 Lecture 20

  12. Small Sample Test for Two Means, Unequal Variances • Test statistic • Degrees of freedom STA 291 Summer 2010 Lecture 20

  13. Small Sample Confidence Interval for Two Means, Unequal Variances STA 291 Summer 2010 Lecture 20

  14. Method 1 (Equal Variances) vs. Method 2 (Unequal Variances) • How to choose between Method 1 and Method 2? • Method 2 is always safer to use • Definitely use Method 2 • If one standard deviation is at least twice the other • If the standard deviation is larger for the sample with the smaller sample size • Usually, both methods yield similar conclusions STA 291 Summer 2010 Lecture 20

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