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STA 291 Fall 2009

STA 291 Fall 2009. Lecture 23 Dustin Lueker. Comparing Dependent Samples. The sample mean of the difference scores is an estimator for the difference between the population means We can now use exactly the same methods as for one sample Replace X i by D i. Comparing Dependent Samples.

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STA 291 Fall 2009

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  1. STA 291Fall 2009 Lecture 23 Dustin Lueker

  2. Comparing Dependent Samples • The sample mean of the difference scores is an estimator for the difference between the population means • We can now use exactly the same methods as for one sample • Replace Xi by Di STA 291 Fall 2009 Lecture 23

  3. Comparing Dependent Samples • Small sample confidence interval Note: • When n is large (greater than 30), we can use the z-scores instead of the t-scores STA 291 Fall 2009 Lecture 23

  4. Comparing Dependent Samples • Small sample test statistic for testing difference in the population means • For small n, use the t-distribution with df=n-1 • For large n, use the normal distribution instead (z value) STA 291 Fall 2009 Lecture 23

  5. Reducing Variability • Variability in the difference scores may be less than the variability in the original scores • This happens when the scores in the two samples are strongly associated • Subjects who score high before the intensive training also dent to score high after the intensive training • Thus these high scores aren’t raising the variability for each individual sample STA 291 Fall 2009 Lecture 23

  6. Example • If we wanted to examine the improvement students made after taking a class we would hope to see what type of value for ? Assuming we take X1-X2 with X1 being the student’s first exam score. • Positive • Negative STA 291 Fall 2009 Lecture 23

  7. Example • Assuming we match people of similar health into 2 groups and gave group 1 a cholesterol medication and measured each groups cholesterol level after 8 weeks, what would we hope would be if we are subtracting group 2 from group 1? • Positive • Negative • Zero STA 291 Fall 2009 Lecture 23

  8. One-Sided Confidence Intervals • Thus far we have only calculated two-sided confidence intervals • A one-sided confidence interval is calculated using nearly the same formula except that we use Zα instead of Zα/2 or tα instead of tα/2 • Also, only one value needs to be calculated as the other value will be the upper or lower bound of the possible values our parameter can obtain • What would these bounds be when estimating a population proportion? STA 291 Fall 2009 Lecture 23

  9. Example • Assume we would like to know the lower bound on the proportion of people who support a particular issue, a sample of 31 individuals reveals that 25 support the issue • Create a one-sided 95% confidence interval for the proportion of people that support the issue • This is sometimes called a 95% lower confidence interval STA 291 Fall 2009 Lecture 23

  10. Correlation Between Tests and Confidence Intervals • Results of confidence intervals and significance tests are consistent • Whenever the hypothesized parameter is not in the confidence interval around the sample statistic, then the p-value for testing H0: μ=μ0 is smaller than 5% (significance at the 5% level) • Why does this make sense? • In general, a 100(1-α)% two-sided confidence interval corresponds to a two-sided test at significance level α • Also, a 100(1-α)% one-sided confidence interval corresponds to a one-sided test at significance level α STA 291 Fall 2009 Lecture 23

  11. Example • A survey of 35 cars that just left their metered parking spaces produced a mean of 18 minutes remaining on the meter and a standard deviation of 22. Test the parking control officer’s claim that the average time left on meters is equal to 15 minutes. • State and test the hypotheses with a level of significance of 5% using the confidence interval method. STA 291 Fall 2009 Lecture 23

  12. Example • A start-up company is about to market a new computer printer. The company hopes a Super Bowl ad will increase the companies name recognition. The goal is that over 40% of the public recognize its brand name and associate it with computer equipment. The day after the Super Bowl a pollster finds that out of 420 people sampled 181 know that this company manufactures printers. Was the ad a success? • Use the confidence interval method to conduct this hypothesis test STA 291 Fall 2009 Lecture 23

  13. Example • Which of these are possible one-sided confidence intervals for a proportion p? • (0, .57) • (.23, 1.06) • (.48, 1) • (.52, .94) STA 291 Fall 2009 Lecture 23

  14. Example • Which of these are possible one-sided confidence intervals for µ, the average life of a light bulb in hours? • (300, 750) • (0, 600) • (150, ∞) • (-200, 600) STA 291 Fall 2009 Lecture 23

  15. Example • A 95% confidence interval for µ is (96,110). Which of the following statements about significance tests for the same data is true? • When testing H1: μ≠100, p-value>.05 • When testing H1: μ≠100, p-value<.05 STA 291 Fall 2009 Lecture 23

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