Example continued 2 Example continued Based on sample
Example continued 2
Example continued Based on sample data, we are 95% confident that the true mean increase in %Fat associated with a year of age is between 0. 320% and 0. 772%. 3
Example continued Minitab output looks like The regression equation is % Fat y = 3. 22 + 0. 548 Age (x) Predictor Constant Age (x) S = 5. 754 Coef 3. 221 0. 5480 SE Coef 5. 076 0. 1056 R-Sq = 62. 7% T 0. 63 5. 19 P 0. 535 0. 000 R-Sq(adj) = 60. 4% Analysis of Variance Source Regression Residual Error Total 4 DF 1 16 17 SS 891. 87 529. 66 1421. 54 MS 891. 87 33. 10 F 26. 94 P 0. 000
Hypothesis Tests Concerning b Null hypothesis: H 0: b = hypothesized value 5
Hypothesis Tests Concerning b Alternate hypothesis and finding the P-value: 1. Ha: b > hypothesized value P-value = Area under the t curve with n - 2 degrees of freedom to the right of the calculated t 2. Ha: b < hypothesized value P-value = Area under the t curve with n - 2 degrees of freedom to the left of the calculated t 3. Ha: b hypothesized value 6 a) If t is positive, P-value = 2(Area under the t curve with n - 2 degrees of freedom to the right of the calculated t) b) If t is negative, P-value = 2(Area under the t curve with n - 2 degrees of freedom to the left of the calculated t)
Hypothesis Tests Concerning b Assumptions: 1. The distribution of e at any particular x value has mean value 0 (me = 0) 2. The standard deviation of e is s, which does not depend on x 3. The distribution of e at any particular x value is normal 4. The random deviations e 1, e 2, … , en associated with different observations are independent of one another 7
Hypothesis Tests Concerning b Quite often the test is performed with the hypotheses H 0: b = 0 vs. Ha: b 0 This particular form of the test is called the model utility test for simple linear regression. The null hypothesis specifies that there is no useful linear relationship between x and y, whereas the alternative hypothesis specifies that there is a useful linear relationship between x and y. 8
Example Consider the following data on percentage unemployment and suicide rates. 9 * Smith, D. (1977) Patterns in Human Geography, Canada: Douglas David and Charles Ltd. , 158.
Example The plot of the data points produced by Minitab follows 10
Example 11
Example Some basic summary statistics 12
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