AP Statistics Section 15 C The most common hypothesis about the slope is _________. H 0 : 0 A regression line with slope 0 is _________. That is, the mean of y (does/does horizontal not) change when x changes. So this says that there is no true linear relationship between x and y. Put another way, says there is ___________ no correlatio n between x and y in the population from which we drew our data. You can use the test for zero slope to test the hypothesis of zero correlation between any two quantitative variables. Note that testing correlation makes sense only if the observations are ______. an SRS The test statistic is just the standardized version of the leastsquares slope b. To test the hypothesis compute the t statistic t b SE b Once again, we use the t-distribution with n - 2 degrees of freedom and as always our p-value is the area under the tail(s). Regressions output from statistical software usually gives t and its two-sided P-value. For a one-sided test, divide the Pvalue in the output by 2. Example 15.6: The hypothesis H : 0 says that crying has no straight-line relationship with IQ. The scatterplot we constructed shows that there is a relationship so it is not surprising that the computer output given on the previous page of notes give t = ______ 3 . 07 with a two-sided P-value of _____. . 004 There is (strong/weak) evidence that IQ is correlated with crying. 0 Example 15.7: A previous example (3.5) looked at how well the number of beers a student drinks predicts his or her blood alcohol content (BAC). Sixteen student volunteers at Ohio State University drank a randomly assigned number of cans of beer. Thirty minutes later, a police officer measured their BAC. Here are the data. Student: 1 2 3 4 5 6 7 8 Beers: 5 2 9 8 3 7 3 5 0.10 0.03 0.19 0.12 0.04 0.095 0.07 0.06 Student: 9 10 11 12 13 14 15 16 Beers: 3 5 4 6 5 7 1 4 0.02 0.05 0.07 0.10 0.085 0.09 0.04 0.05 BAC: BAC: Here is the Minitab output for the blood alcohol content data: The regression equation is BAC = - 0.0127 + 0.0180 Beers Predictor Coef StDev T P Constant -0.01270 0.01264 -1.00 0.332 Beers 0.017964 0.002402 7.48 0.000 S = 0.02044 R-Sq = 80.0% Note: the actual calculated values are slightly different from these Test the hypothesis that the number of beers has no effect on BAC. Hypotheses: The population of interest is __________________ college students OSU H0: _______ 0 In words, ____________________________ the # of beers has no effect on BAC H1: _______ 0 In words, _________________________________ the # of beers has a positive effect on BAC where is ___________________________ the slope of the true regression line Conditions: Seems safe to assume we have an SRS and responses are independen t. Scatterplo t and residual plot indicate a linear relationsh ip Residual plot shows constant standard deviation of the responses about LSL Analysis shows approximat ly Normal distributi on to for the residuals Calculations: P - value .000 7.48 t 7 . 48 NOTE : t b SE b . 017964 . 002402 7 . 48 Interpretation: At any common significan ce level we reject H 0 and conclude that an increase in the number of beers will cause an increase in BAC.