Interval half width LCL and UPL 145 TESTING THE SIGNIFICANCE OF THE SLOPE OF

Interval half width lcl and upl 145 testing the

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Interval half width - LCL and UPL 14.5 – TESTING THE SIGNIFICANCE OF THE SLOPE OF THE REGRESSION EQUATION o Interpreting the Meaning of Slope = 0 For any value of x, the predicted value of y will always equal the y-intercept No relationship b/w the dependent and independent variables Perform a hypothesis test to det. if the population regression slope (M) is significantly diff from zero based on the sample regression slope (m) The Hypothesis Test o Step 1: The hypotheses: - H0: M = 0 - H1: M ≠ 0 If reject null hypothesis - There is a significant relationship b/w the dependent and independent variables o Step 2: T Test Statistic T = (m – M) / Sb - m = sample regression slope - M = population regression slope (from null hypothesis) - Sb = standard error of the slope In our example = 2.96 o Step 3. Standard Error of the Slope (Sb) – measures how consistent the slope of the regression equation (b) would be if several sets of samples from the population were selected and the regression equation were derived for each of them
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Sb = (Se) ÷ √ ∑x² – n(sample mean of x)² Regression equation calculated for many diff samples Det. how close the slopes for these samples are to one another Small Sb – increases likelihood that can est. significant relationship b/w the two variables Large Sb - low likelihood that can est. significant relationship b/w the two variables In our example = 1.41 o Step 4. Critical T-Score DF = (n – 2) = 4 ∞ = 0.05 Two-tail → 2.776 o Step 5. Comparing T and the T-Score 2.96 (T) > 2.776 (T-Score) → REJECT o Step 6: State your conslusions Reject null hypothesis The population regression slope (M) is NOT equal to zero A relationship exists b/w the hours studied and exam grades CONFIDENCE Interval for M o Formula – Confidence Interval for the Regression Slope CI = m ± (t-score • Sb) Critical T-Score - ∞ = 0.05 of 95% condience interval at DF (n – 2) = 4 - → 2.776 In our example, CI = 4.1786 ± 3.914 - UCL = 8.093 - LCL = 0.265 o Results: 95% confident that true population slope is b/w
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- 0.265 and 8.093 Every additional hour of study (one more x) will increase your score by 0.265 – 8.093 points Confidence interval does NOT include zero - Evidence to conclude that there is a relationship b/w the variables o Finding it in Excel Se = under Regression Statistics T-Score in “t Stat” column, 2 nd row LCL and UCL in “lower 95%” column, 2 nd row P-value in P-value column, 2 nd row o P-Value in Excel Represents area to the right and left of the extreme ± values of the T-Score If P < ∞ → REJECT When p-value and t-test are identical - b/c only have one independent variable 14.6 – ASSUMPTIONS FOR REGRESSION ANALYSIS o The Basics Key assumptions need to hold true for regression analysis results to be reliable o Assumption: the relationship b/w the independent and dependent variables is linear Non-linear pattern affects regression results - Some predicted Y values may end up in negative o Assumption 1: Residuals
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  • Fall '12
  • Donnelly
  • Normal Distribution, Null hypothesis, Hypothesis testing, Statistical hypothesis testing

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