Elementary Statistics: Interval Estimates for Two Quantitative Variables, Study notes of Statistics

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(C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture 1 (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture Lecture 35 Two Quantitative Variables Interval Estimates PI for Individual Response, CI for Mean Response Explanatory Value Close to or Far from Mean Approximating Intervals by Hand Width of PI vs. CI Guidelines for Regression Inference (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.2 Looking Back: Review  4 Stages of Statistics  Data Production (discussed in Lectures 1-4)  Displaying and Summarizing (Lectures 5-12)  Probability (discussed in Lectures 13-20)  Statistical Inference  1 categorical (discussed in Lectures 21-23)  1 quantitative (discussed in Lectures 24-27)  cat and quan: paired, 2-sample, several-sample (Lectures 28-31)  2 categorical (discussed in Lectures 32-33)  2 quantitative (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.3 Correlation and Regression (Review)  Relationship between 2 quantitative variables  Display with scatterplot  Summarize:  Form: linear or curved  Direction: positive or negative  Strength: strong, moderate, weak If form is linear, correlation r tells direction and strength. Also, equation of least squares regression line lets us predict a response for any explanatory value x. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.4 Population Model; Parameters and Estimates Summarize linear relationship between sampled x and y values with line minimizing sum of squared residuals . Typical residual size is Model for population relationship is and responses vary normally with standard deviation  Use to estimate  Use to estimate  Use to estimate (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture 2 (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.5 Regression Null Hypothesis (Review)  no population relationship between x and y Test statistic t P-value is probability of t this extreme, if true (where t has n-2 df) (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.6 Confidence Interval for Slope (Review) Confidence interval for is where multiplier is from t dist. with n-2 df. If n is large, 95% confidence interval is If CI does not contain 0, reject , conclude x and y are related. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.7 Interval Estimates in Regression Seek Prediction and Confidence Intervals for  Individual response to given x value (PI)  For large n, approx. 95% PI:  Mean response to subpopulation with given x value (CI)  For large n, approx. 95% CI: Both intervals centered at predicted y-value These approximations may be poor if n is small or if given x value is far from average x value. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.8 Example: An Interval Estimate  Background: Property owner thinks reassessed value $40,000 of his 4,000 sq.ft. lot is too high. Sizes for random sample of 29 local lots have mean 5,619 sq.ft.; values have mean $34,624. r=+0.927, regression equation , s=$6,682.  Question: Is there evidence that his value is significantly higher than usual? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture 5 (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.22 Example: Estimate Individual Wt, No Ht Info  Background: A sample of male weights have mean 170.8, standard deviation 33.1. Shape of distribution is close to normal.  Response: Need to know distribution of weights is approximately normal to apply 68-95-99.7 Rule: Approx. 95% of individual male weights in interval _________________________________________ (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.24 Examples: Series of Estimation Problems  Based on sample of male weights, estimate  weight of individual male  mean weight of all males  Based on sample of male heights and weights, est  weight of individual male, 71 inches tall  mean weight of all 71-inch-tall males  weight of individual male, 76 inches tall  mean weight of all 76-inch-tall males (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.25 Example: Estimate Mean Wt, No Ht Info  Background: A sample of 162 male weights have mean 170.8, standard deviation 33.1.  Questions:  What interval should contain the mean weight of all males?  How does it compare to the interval for an individual male’s weight? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.27 Example: Estimate Mean Wt, No Ht Info  Background: A sample of 162 male weights have mean 170.8, standard deviation 33.1.  Responses:  Need to know sample size n to construct approximate 95% confidence interval for mean: ___________________________________________  Interval for mean involves division by square root of n __________ than interval for individual. (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture 6 (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.29 Examples: Series of Estimation Problems  Based on sample of male weights, estimate  weight of individual male  mean weight of all males  Based on sample of male heights and weights, est  weight of individual male, 71 inches tall  mean weight of all 71-inch-tall males  weight of individual male, 76 inches tall  mean weight of all 76-inch-tall males Need regression (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.30 Examples: Series of Estimation Problems (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.31 Example: Find Individual Wt, Given Average Ht  Background: Regression of male weight on height has r =+0.45, p=0.000strong evidence of moderate positive relationship. Reg. line and s=29.6 lbs ( lbs); mean ht. about 71.  Questions:  How much heavier is a sampled male, for each additional inch in height?  Why is ?  What interval should contain the weight of an individual 71-inch-tall male? (Got interval estimates for x=71.) (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.33 Example: Find Individual Wt, Given Average Ht  Background: Regression of male weight on height has r =+0.45, p=0.000strong evidence of moderate positive relationship. Reg. line and s=29.6 lbs ( lbs); mean ht. about 71  Responses:  For each additional inch in height, a male weighs about ___ lbs more (slope).  because wts vary ___ about line than about mean  Software___ for x=71: (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture 7 (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.34 Example: Est Individual Wt, Given Average Ht  Background: Regression of male weight on height has r =+0.45, p=0.000strong evidence of moderate positive relationship. Reg. line and s=29.6 lbs. Got interval estimates for x=71.  Questions:  How do we approximate interval estimate for wt. of an individual 71-inch-tall male by hand?  Is our approximate close to the true interval? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.36 Example: Est Individual Wt, Given Average Ht  Background: Regression of male weight on height has r =+0.45, p=0.000strong evidence of moderate positive relationship. Reg. line and s=29.6. Got interval estimates for x=71.  Responses:  Predict y for x=71: PI=______________________________________  Close? ____ (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.38 Examples: Series of Estimation Problems  Based on sample of male weights, estimate  weight of individual male  mean weight of all males  Based on sample of male heights and weights, est  weight of individual male, 71 inches tall  mean weight of all 71-inch-tall males  weight of individual male, 76 inches tall  mean weight of all 76-inch-tall males (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.39 Example: Est Mean Wt, Given Average Ht  Background: Regression of 162 male wts on hts has r =+0.45, p=0.000strong evidence of moderate positive relationship. Reg. line and s=29.6 lbs. Got interval estimates for x=71.  Questions:  What interval should contain mean weight of all 71-inch- tall males?  How do we approximate the interval by hand? Is it close? (Need to know sample size to get margin of error for mean) (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture 10 (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.53 Example: Est Individual Wt, Given Tall Ht  Background: Regression of male weight on height has r =+0.45, p=0.000strong evidence of moderate positive relationship. Reg. line and s=29.6 lbs. Got interval estimates for x=76.  Responses:  Predict y for x=76: ______________________________ PI=______________________________________________  Close? ____ (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.55 Examples: Series of Estimation Problems  Based on sample of male weights, estimate  weight of individual male  mean weight of all males  Based on sample of male heights and weights, est  weight of individual male, 71 inches tall  mean weight of all 71-inch-tall males  weight of individual male, 76 inches tall  mean weight of all 76-inch-tall males (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.56 Example: Est Mean Wt, Given Tall Ht  Background: Regression of 162 male wts on hts has r =+0.45, p=0.000strong evidence of moderate positive relationship. Reg. line and s=29.6 lbs. Got interval estimates for x=76.  Questions:  What interval should contain mean weight of all males who are 76 inches tall?  How do we approximate the interval by hand? Is it close? (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.58 Example: Est Mean Wt, Given Tall Ht  Background: Regression of 162 male wts on hts has r =+0.45, p=0.000strong evidence of moderate positive relationship. Reg. line and s=29.6 lbs. Got interval estimates for x=76.  Responses:  Refer to ___  Predict y for x=76:_______________________________ CI=______________________________________________ Close? __________ (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture 11 (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.59 Examples: Series of Estimation Problems PI CI * * (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.60 Interval Estimates in Regression (Review) Seek interval estimates for  Individual response to given x value (PI)  For large n, approx. 95% PI:  Mean response to subpopulation with given x value (CI)  For large n, approx. 95% CI:  Intervals approximately correct only for x values close to mean; otherwise wider  Especially CI much wider for x far from mean (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.61 PI and CI for x Close to or Far From Mean : (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.63 Summary of Example Intervals : CI alwaysnarrower than PI (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture 12 (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.64 Summary of Example Intervals : CI and PI can be narrower if x info given (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.65 Summary of Example Intervals : CI and PI centered at heavier wt for taller ht (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.66 Summary of Example Intervals : CI and PI wider for wt far from average (C) 2007 Nancy Pfenning Elementary Statistics: Looking at the Big Picture L35.67 Guidelines for Regression Inference  Relationship must be linear  Need random sample of independent observations  Sample size must be large enough to offset non- normality  Need population at least 10 times sample size  Constant spread about regression line  Outliers/influential observations may impact results  Confounding variables should be separated out