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← Confidence Intervals - Intro to Inferential Statistics

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Showing Revision 5 created 05/25/2016 by Udacity Robot.

  1. Although using the standard error of the estimate can help us assess the
  2. accuracy of our predictions. We can make even more accurate predictions by
  3. computing confidence intervals for our predicted values, our y-hats. In other
  4. words, when we get our regression line, we have our expected value which is
  5. this y-hat naught, for a specified value x-naught. However, the actual value
  6. might be anything from up here, to down here, or it might be exactly equal to
  7. our predicted value. Therefore, we might want a confidence interval around our
  8. expected value for where the actual value might be. Similarly, we might want a
  9. confidence interval for the true slope. We'll have a certain regression line
  10. and we'll have calculated the slope for that regression line based on our
  11. sample data. And this is just an example, the slope could be this, or flatter
  12. down to here. A confidence interval for a slope can tell us the range for which
  13. the true population's slope might be. You won't actually calculate the
  14. confidence interval in this lesson, because we're going to assume that you'll
  15. use a computer to get it. The important thing is that you know what it means.
  16. For example, let's say we have some sample data that looks like this. We
  17. calculate that the regression line is y equals bx plus a. Some slope and some
  18. intercept. But lets say then that we're able to look at all the population
  19. data. And it looks like this. Now it looks to be slightly downward sloping.
  20. Since we're assuming this is the true regression line, we'll use the common
  21. notation for the regression coefficients for the population. Oh, and these
  22. should have hats since they're the predicted values. If this were the case,
  23. where the sample regression line is positively sloping. But the true regression
  24. line for the population is negatively sloping. That would mean that the
  25. confidence interval for b has a negative lower bound and a positive upper
  26. bound. It includes zero within this range. Therefore, if we run a two-tailed
  27. hypothesis test for whether or not the slope is equal to zero. We would fail to
  28. reject the null, meaning there's no evidence that there's a linear relationship
  29. between x and y based on that sample. In fact, let's get more into hypothesis
  30. testing for slope.