# Zero appears to be a relationship bw the two

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zero – appears to be a relationship b/w the two variablesoIn the REGRESSION OUTPUT – EXCELRegression Statistics > R Square-= coefficient of determination, R²Sum of squares (ANOVA – SS column)-Residual = error sum of squares
-RegressionANOVA F column = F-scoreANOVA DF shows degrees of freedom for-D1 (regression)-D2 (residual)ANOVA Significance F = p-value-Represents area under distribution to the right and/or left ofthe calculated test statistic (F)oEvaluating the ResultsP value < ∞ → REJECTP value > ∞ → DON’T REJECT14.4 – USING A REGRESSION TO MAKE A PREDICTIONoThe BasicsHow reliable is our estimate (based on y = mx + b)-Only have info on sample – need populationConfidence interval around predicted exam grade to det. accuracy of the regression lineoStandard Error of the Estimate– (Se) measures the amount of dispersion of observed data around a regression line= √ SSE / (n – 2)Low (Se) – data points are close to lineLarge (Se) – data points are farther from the regression lineIn our example = 4.32The Confidence Interval for an Average Value of Y Based on a Value of XExample: Confidence interval for the average exam score for students that studied 3 hoursoFormula – Confidence Interval (CI) for an Average Value of Y= y ± (T • Se)√ 1/n + [ (x – avg value of x)² ÷ ∑x² - (∑x)²/n-Y = predicted value for y given x = __-T = T Test Statistic
-Se = standard error of the estimate-N = number of ordered pairs-X = the value of xoThe ProcessAverage Value of X-= ∑x / n-In our example = 3.667T Test Statistic-DF = (n – 2) at two-tail ∞ = 0.05-In our example = 2.776Confidence interval-In our example = 82.7 ± 5.55-UCL = 82.7 + 5.55 = 88.3-LCL = 82.7 – 5.55 = 77.2oEvaluating95% confident that average exam grade (average value of y) earned by students who studied 3 hours (at x = 3) will be between -77.2 and 88.3Width of confidence interval-= 88.3 – 77.2 = 11.1Interval is narrowest at average value of x-(avg. value of x) = ∑x / n = 22/6 = 3.67As get further away from the mean, intervals widen-Estimates are more precise the closer we are to the meanThe Prediction Interval for a Specific Value of Y Based on a Value of XoThe basicsPrediction interval for a single Y (grade) given X (hours studied)-Ex: predict my grade given that I studied 3 hoursoFormula – Prediction Interval (PI) for a Specific Value of Y= y ± (T • Se)√ 1/n + [1 + (x – avg value of x)² ÷ ∑x² - (∑x)²/n ]
Same as CI, but includes (1 + ____) in the beginning of the square rootoEx: 95% prediction interval if study for 3.0 hoursPI = 82.7 ± 13.21UPL = 82.7 + 13.21 = 95.9LCL = 82.7 – 13.21 = 69.595% confidence interval [69.5, 95.9]oSome NotesPrediction interval variation is greater than if estimated average-b/c estimating individual average, not an average of a data setUsing PHStat2 to Construct Confidence and Prediction IntervalsoProceduresInput variable data into different columnsAdd-Ins > PHStat2 > Regression > Simple Linear RegressionInput data where needed-Y variable range-X variable range-Whether contains labels-Confidence level (ex: 95%)CHECK very bottom box before click OK-Confidence and prediction interval for x = ___ (insert)-Confidence level for interval estimates ___ (insert %)o
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