Idea Transcript
Chapter 12: Inference for Linear Regression Confidence Interval for Slope We are going to learn how to construct a confidence interval to estimate the unknown slope β of the population (true) regression line because that is usually the most important parameter in the regression problem. Confidence Interval Example: A study investigated why some people don’t gain weight even when they overeat. Perhaps fidgeting and other “nonexercise activity” (NEA) explains why – some people may spontaneously increase nonexercise activity when fed more. Researchers deliberately overfed a random sample of 16 healthy young adults for 8 weeks. They measured fat gain (in kilograms) and change in energy use (in calories) from activity other than deliberate exercise – fidgeting, daily living, and the like – for each subject. Here are the data and the Minitab output from a least-squares regression analysis for these data. NEA Change (cal) Fat Gain (kg)
-94 4.2
-57 3.0
-29 3.7
135 2.7
143 3.2
151 3.6
245 2.4
355 1.3
392 3.8
473 1.7
486 1.6
535 2.2
571 1.0
580 0.4
620 2.3
690 1.1
Construct and interpret a 90% confidence interval for the slope of the population regression line Step 1: State the name of the confidence interval and the parameter of interest that we are estimating. We are interested in the slope β of the true regression line that says how much fat gain occurs for each calorie burned through NEA. Step 2: Conditions for Regression Inference (Confidence Interval and Hypothesis Test) Linear – The actual relationship between x and y is Linear. For any fixed value of x, the mean response 𝜇! falls on the population (true) regression line 𝜇! = 𝛼 + 𝛽𝑥 The slope β and intercept α are unknown parameters. Independent – Individual observations are Independent of each other. Normal – For any fixed value of x, the response y varies according to a Normal distribution. In other words the residuals should have an approximately normal distribution. A stemplot of the residuals will show this. Equal variance – The standard deviation of y (call it σ) is the same for all values of x. In other words, the residuals have Equal variance. A residual plot will show this – We want the residuals to appear random and we do not want the residuals to follow a pattern (growing, shrinking, etc.). The common standard deviation σ is usually an unknown parameter. Random – The data comes from a well-designed Random sample or a randomized experiment. ********************Remember L.I.N.E.R.*********************
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Step 3: Calculations From the AP stat packet:
statistic ± (critical value) · (standard deviation of statistic)
For slope, this formula becomes:
b ± t ∗ SE b
Regr essio n Analysis: F at gain v ers us NEA c hang e Predictor
Coef
SE Coef
T
P
Constant
3.5051
NEA Change
-0.0034415
0.3036
11.54
0.000
0.0007414
-4.64
0.000
S = 0.739853
R-Sq = 60.6%
R-Sq(adj) = 57.8%
The statistic b is -.0034. For the standard deviation of slope, look in the SE Coef. column next to the variable to be multiplied by the slope. In the example, this is NEA Change and the standard deviation is .0007414. 𝑡 has (n - 2) degrees of freedom. There are 16 ordered pairs in the table, so there are 14 DF. With .90 between the tails, the T-Value is 1.76. IN AP STAT PROGRAM: Run Linear Regression, then Error Analysis Error: 1.76 • .0007414 = .0013
Interval: (-.00475, -.00214)
Step 4: Interpretation We are 90% confident that the slope of the true regression line is captured by the interval: (-.00475, -.00214). NOTE: the variable S in the printout above is the standard error of the residuals. This is the standard deviation of the residuals, which tells the average distance away from the line the data values fall.
Interpreting the computer printout:
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Activity: Helicopters! Make 25 Helicopters. Select 5 drop heights (recommended: drop from over 5 ft). Drop 5 from each height and record the time it takes to hit the ground. Hght 1:
Hght 2:
Hght 3:
Hght 4:
Hght 5:
Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Make a scatterplot for the data:
Find the Least Squares Regression Equation: Find a 95% Confidence Interval for the Slope: HW A: 1, 3 3
Day 2: Confidence Intervals for Slope Opener – Review: State the checks necessary for inference:
Example 1: The following is the output for data that compared the performance of a golfer in the first and second round of a tournament. The data are in the table below: Golfer 1 Round1 89 Round2 94
2 90 85
3 87 89
4 95 89
5 86 81
6 81 76
7 102 107
8 105 89
9 83 87
10 88 91
11 91 88
12 79 80
a.) What is the regression equation for this data (𝑦 = 𝑎 + 𝑏𝑥)?
b.) What is the standard error about the line (s)?
c.) What is the standard error of the slope (SEb)?
d.) Construct a 95% confidence interval for the slope of the true regression line.
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Example 2: Do Longer Drives Equal Lower Scores?
Find the LSRL and explain the slope in context: State the correlation and coefficient of determination. Explain the meaning of r2: What is the standard error of the line? What is the standard error of the slope? Perform the checks for inference: State a 95% confidence interval for the slope (show all necessary calculations):
Conclusion:
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Example 3:
HW B: 5, 7, 9, 11 6
Day 3: Significance Tests for Slope
The null hypothesis will be The alternate hypothesis will be
H0: slope = 0 This means that changing x has no effect on y Ha: slope > 0 slope < 0 or slope ≠0
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Example 1
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HW C: 13, 15, 17, 19, 21-26 9
AP Exam Format Students are not expected to memorize any formulas; rather, a list of common statistical formulas related to descriptive statistics, probability, and inferential statistics is provided. Moreover, tables for the normal, Student's t and chi-squared distributions are given as well. Students are also expected to use graphing calculators with statistical capabilities. The exam is three hours long with ninety minutes allotted to complete each of its two sections: multiple choice and free-response. The multiple choice portion of the exam lasts 90 minutes and consists of forty questions with five possible answers each. The free response section lasts 90 minutes and contains six open-ended questions that are often long and divided into multiple parts. The first five of these questions may require twelve minutes each to answer and normally relate to one topic or category. The sixth question consists of a broad-ranging investigative task and may require approximately twenty-five minutes to answer.
How we will study: Each day you will complete ¼ of a practice test: Day 1: questions 1-20 from MC Day 2: questions 21-40 from MC Day 3: questions 1-4 from FR Day 4: question 5 from FR and question 6 – Investigative Task Day 5: repeat with new practice exam First practice test is the Cummulative AP Review on pages 799 to 809. Note: You must time yourself to see if you are answering questions at the correct pace. Set the timer for 45 minutes. Mark on your practice test where you were when time ran out THEN finish the assignment for the day. 10