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Confidence Intervals for the Regression Slope

12.1b Target Goal: I can perform a significance test about the slope β of a population (true) regression line. h.w: Pg 761: 13 - 19 odd. Confidence Intervals for the Regression Slope. Review Linear Regression. Weight. Height.

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Confidence Intervals for the Regression Slope

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  1. 12.1b Target Goal: I can perform a significance test about the slope β of a population (true) regression line. h.w: Pg 761: 13 - 19 odd Confidence Intervals for the Regression Slope

  2. Review Linear Regression

  3. Weight Height Let’s look at the heights and weights of a population of adult men. How much would an adult male weigh if he were 5 feet 4 inches tall? Weights of men will vary – in other words, there is a distribution of weights for adult males who are 5 feet 4 inches tall. Are some of these weights more likely than others? What would this distribution look like? We want the standard deviations of all these normal distributions to be the same. Where would you expect the population regression line to be? What would you expect for other heights? This distribution is normally distributed. 70 72

  4. The Sampling Distribution of b : Old Faithful cont. Inference for Linear Regression Let’s return to our earlier exploration of Old Faithful eruptions. For all 222 eruptions in a single month, the population regression line for predicting the interval of time until the next eruption y from the duration of the previous eruption x is µy= 33.97 + 10.36x. The standard deviation of responses about this line is given by σ = 6.159. If we take all possible SRSs of 20 eruptions from the population, we get the actual sampling distribution of b. Shape: Approx. Normal µb = β = 10.36 Center : (b is an unbiased estimator of β) In practice, we don’t know σ for the population regression line. So we estimate it with the standard deviation of the residuals, s. Then we estimate the spread of the sampling distribution of b with the standard error of the slope:

  5. Suppose the conditions for inference are met. To test the hypothesis H0: β = hypothesized value, compute the test statistic Find the P-value by calculating the probability of getting a t statistic this large or larger in the direction specified by the alternative hypothesis Ha. Use the t distribution with df = n - 2. • Performing a Significance Test for the Slope Inference for Linear Regression Typically = 0

  6. Example: Crying and IQ Inference for Linear Regression Infants who cry easily may be more easily stimulated than others. This may be a sign of higher IQ. Child development researchers explored the relationship between the crying of infants 4 to 10 days old and their later IQ test scores. A snap of a rubber band on the sole of the foot caused the infants to cry. The researchers recorded the crying and measured its intensity by the number of peaks in the most active 20 seconds. They later measured the children’s IQ at age three years using the Stanford-Binet IQ test. A scatterplot and Minitab output for the data from a random sample of 38 infants is below. Do these data provide convincing evidence that there is a positive linear relationship between crying counts and IQ in the population of infants?

  7. Example: Crying and IQ Inference for Linear Regression Do these data provide convincing evidence that there is a positive linear relationship between crying counts and IQ in the population of infants? State: We want to perform a test of H0: β = 0 Ha: β > 0 where β is the true slope of the population regression line relating crying count to IQ score. No significance level was given, so we’ll use α = 0.05.

  8. Example: Crying and IQ Inference for Linear Regression Plan: If the conditions are met, we will perform a t test for the slope β. • Linear:The scatterplot suggests a moderately weak positive linear relationship between crying peaks and IQ. The residual plot shows a random scatter of points about the residual = 0 line. • Independent: Later IQ scores of individual infants should be independent. Due to sampling without replacement, there have to be at least 10(38) = 380infants in the population from which these children were selected.

  9. Example: Crying and IQ Inference for Linear Regression • Normal: The Normal probability plot of the residuals shows a slight curvature, which suggests that the responses may not be Normally distributed about the line at each x-value. With such a large sample size (n = 38), however, the t procedures are robust against departures from Normality. • Equal variance: The residual plot shows a fairly equal amount of scatter around the horizontal line at 0 for all x-values. • Random:We are told that these 38 infants were randomly selected.

  10. Example: Crying and IQ Inference for Linear Regression Do: With no obvious violations of the conditions, we proceed to inference. The test statistic and P-value can be found in the Minitab output. The Minitab output gives P = 0.004 as the P-value for a two-sided test. The P-value for the one-sided test is half of this, P = 0.002. Conclude: The P-value, 0.002, is less than our α = 0.05 significance level, so we have enough evidence to reject H0and conclude that there is a positive linear relationship between intensity of crying and IQ scorein the population of infants.

  11. Testing the Hypothesis of No Linear Relationship • To test whether or not there is a correlation between two quantitative variables, consider the slope of the regression line. • If there is no correlation, the slope would be zero. H0: β = 0 (The mean of y does not change at all as x changes.)

  12. To test this hypothesis, compute the t statistic and P-value. Note: • Regression output from statistical software usually gives t and its two-sided P-value. • For a one-sided test,divide the P-value in the output by 2.

  13. Ex: Beer and Blood Alcohol • How well does the number of beers a student drinks predict his or her blood alcohol content? • Sixteen of age college student volunteers drank a randomly assigned number of cans of beer. Thirty minutes later, a police officer measured their blood alcohol content (BAC).

  14. Data revealed: • They noticed a variation in the data and didn’t believe that the number of drinks predicted the BAC well. Here is the scatter plot. • The solid line is the LSRL. • The scatterplot shows a clear linear relationship.

  15. Minitab output: Because r2 = 0.8000, the number of drinks accounts for80% of the observed variation in BAC. (The students are wrong).

  16. To test hypothesis that the number of beers has no effect on BAC; Ho: β = 0 There is no correlation between the number of beers consumed and BAC. Ha: β > 0 There is a positive correlation between the number of beers consumed greater the BAC. • Examine the P-value • From the output: P-value = 0.0000, • The one sided P-value is half of this so it is also close to 0.

  17. Thus, we reject Ho and conclude that the number of beers does have an effect on BAC. • Calculator (page 804): LinRegTest • (Do for “crying vs. IQ.)

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