Prediction Intervals and Confidence Intervals

Transcription

1 Four Stages of Statistics Prediction Intervals and Confidence Intervals Lecture 32 April 13, 2018 Data Collection Displaying and Summarizing Data Probability Inference One Quantitative One Categorical One Categorical and One Quantitative Two Categorical Two Quantitative Simple Linear Regression Multiple Linear Regression Comparing Types of Error Two types of variation in regression: explained and unexplained Explained Variation:differences in the responses due to the relationship between the predictor and response () Unexplained Variation:differences in the responses due to natural variability in the population () Related to distance a point is from regression line Standard Error of the Model Standard Error of the Model:standard deviation of the error term = 2 Measure of how far away observations are from the regression line Average residual Large values of imply many points are far away, indicating some large residuals Small values of imply most points are close to the regression line, indicating a good fit

2 Example #1: Standard Error of the Model. Question: What is the model s standard error? Example #1: Standard Error of the Model. Question: What does the standard error mean? A typical admission price Two Types of Intervals Two types of intervals exist for performing inference in regression. Choose a value from predictor variable: Prediction Interval:range of reasonable values for where an individual response at the selected value of the predictor will fall Confidence Interval:range of reasonable values for where the mean of all responses at the selected value of the predictor will fall Prediction Interval Prediction Interval:an interval estimate used to predict where an individual response will fall at a specific value of the predictor ± is the best estimate for the next observation at Prediction interval gives a range of reasonable values for the where the response of an individual observation taken at is likely to fall

3 Confidence Interval Confidence Interval:an interval estimate used to estimate the expected value of at a specific value of the predictor 1 ± + 1 is the best estimate for the mean of all observations at Confidence interval gives a range of reasonable values for the mean of all responses at Four Types of Inference Variable Mean Standard Deviation Price() Rides() Question: How can we perform inference on 1. Admission price for a single park with 37 rides? 2. Average admission price for parks with 37 rides? 3. Admission price for a single park with 20 rides? 4. Average admission price for parks with 20 rides? Four Types of Inference = =35 = Note: =2.035 Example #2: Point Estimates Question:What is the point estimate for the admission price for a park with 37 rides? Question:What is the point estimate for the admission price for a park with 20 rides?

4 Example #3: Predict Observation at =37 Question:What interval approximates the admission price for a single park with 37 rides? Example #4: Approximate Mean at =37 Question:What interval approximates the mean admission price for all parks with 37 rides? Example #5: Predict Observation at =20 Question:What interval approximates the admission price for a single park with 20 rides? Example #6: Approximate Mean at =20 Question:What interval approximates the mean admission price for all parks with 20 rides?

5 Conf. Int. vs. Pred. Int. at Same Question:What do you notice about the prediction and confidence intervals at the same value of the predictor? Interval = Prediction (27.57, 76.39) Confidence (47.92, 56.04) Interval = Prediction (12.11, 61.57) Confidence (31.17, 42.51) Conf. Int. vs. Pred. Int. at Same Reason:Prediction intervals are for, while confidence intervals are for CI: + PI: 1+ + Comparing Conf. Int. at Different Question:What do you notice about the difference between the confidence interval at =20compared to at =37? Predictor Interval =37 (47.92,56.04) =20 (31.17,42.51) Width Comparing Pred. Int. at Different Question:What do you notice about the difference between the prediction interval at =20compared to at =37? Predictor Interval =37 (27.57,76.39) =20 (12.11,61.57) Width

6 Comparing Intervals at Different Takeaway:Both prediction and confidence intervals are Reason: Have about values near than those farther away. If is farther from, then becomes larger, which makes the margin of error larger. Example #7: Interpreting Interval Scenario:A 95% prediction interval for the admission price for parks with 37 rides is (27.57, 76.39). Question:What is the correct interpretation of this interval? Answer:We are 95% confident that CI: + PI: 1+ + Example #8: Interpreting Interval Scenario:A 95% confidence interval for the admission price for parks with 37 rides is (47.92, 56.04). Question:What is the correct interpretation of this interval? Answer:We are 95% confident that Example #9: Wider Intervals Scenario:A 95% confidence interval for the admission price for parks with 37 rides is (47.92, 56.04). Question:Which of the following intervals would be wider? I. A 95% prediction interval for admission price for parks with 37 rides II. A 95% confidence interval for admission price for parks with 50 rides III. A 99% confidence interval for admission price for parks with 37 rides Answer:

7 Summary Standard Error:standard deviation of the error term Prediction Interval:interval estimate of the response for a single observation taken at Confidence Interval:interval estimate of the mean of all responses taken at