Lecture #25 Tuesday, November 15, 2016 Textbook: 14.1 and 14.3 Statistics 200 Lecture #25 Tuesday, November 15, 2016 Textbook: 14.1 and 14.3 Objectives: • Formulate null and alternative hypotheses involving regression coefficients. • Calculate T-statistics; determine correct degrees of freedom.

Refresher Chapter 3 covered regression equations and the relationship between two __________ variables. quantitative Example: Q. What is your height in inches? Q. What is your father’s height in inches? Which variable should be the explanatory variable? Father’s height Student’s height

response explanatory Regression equation __________ on the y-axis R-squared ___________ on the x-axis explanatory

Regression terminology The linear regression equation looks like this: Recall that “y-hat” is the estimated mean of the response (y). It can also refer to a predicted value of y. slope Estimated (mean) response explanatory y-intercept

In our example: y-hat = 30.34 + 0.53 x (dadheight) If the dad was 0 inches tall, the predicted student height is 30.34 inches. CAREFUL! y-hat = 30.34 + 0.53 x (dadheight) For every one inch increase in _______, the predicted _______ ___________ by ______ inches. dadheight height increases .53

Refresher An important point to remember is that if the slope (b1) is different from zero, then the two quantitative variables are linearly related.

Parameters for Regression When regression was discussed before, we only talked about it in terms of the sample statistics: _____ (sample intercept) and ___ (sample slope). b0 b1 parameters However, there are also ____________ for the slope and intercept in a linear regression equation. These parameters represent the intercept and slope that would be found if the whole population for both variables was used to create a regression equation.

Parameters for Regression ____ is the population intercept. It is estimated by the sample intercept (b0). ____ is the population slope. It is estimated by the sample slope (b1). β0 β1

Parameters for Regression _____ is the population mean response (i.e. expected value of y for all individuals in the population who have the particular value of x.) Note that ______ is an estimate of ____. The value epsilon is called the error or the deviation. It has a mean of zero. (And we assume it is normally distributed.) E(y) y-hat E(y)

Parameters for Regression If two variables have a linear relationship, then β1 (the population slope) would be different from _____.

Hypothesis Testing About the Slope Statistical significance of a linear relationship can be evaluated by testing whether the population slope is ______ or not. This test is done in a similar way to tests with proportions and means. First, the null and alternative hypotheses need to be determined.

Null and Alternative Hypotheses Null hypothesis H0: ______ This would mean that our two variables, x and y, _______ linearly related Alternative hypothesis Ha: _________ The variables x and y ____ linearly related β1 = 0 are not β1 ≠ 0 are

Null and Alternative Hypotheses 1-sided The alternative hypothesis can be _______ as well (β1 > 0 or β1 < 0), but most software use the _______ alternative hypothesis (β1 ≠ 0) We will only use the ________ alternative hypothesis 2-sided 2-sided

The Test Statistic t-statistic For the hypothesis tests for slope, the __________ is used. The t-statistic is calculated in the same way as before:

The Test Statistic When we are using the t-test for the test of the slope, the degrees of freedom are equal to the sample size minus two. df = ________ n – 2

The Test Statistic The calculations for the sample slope and its standard error are complicated Luckily, Minitab can do this for us: Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 30.34 5.08 5.98 0.000 dadheight 0.5280 0.0732 7.21 0.000 1.00 p-value b1 s.e.(b1) t-stat

Example: Age and Reading Distance A sample was taken in which subjects were asked their age, and then they were measured to see how far away they could read a road sign. Age was treated as the explanatory variable, and reading distance was the response variable. There are n = 30 observations

Example: Age and Reading Distance The sample slope was –3.0068, which means that for each additional year of age, the estimated reading distance ____________ by about 3 feet. The standard error for the slope was 0.4243. decreased

Example: Age and Reading Distance The t-statistic is calculated like this: The Minitab output would look like this:

Example: Age and Reading Distance The correct conclusion is that, since the p-value is ______ , the null hypothesis should be ________. This would mean the slope is significantly __________ from ______. So age and reading distance ___________________. < 0.05 rejected different are linearly related

Confidence Intervals for Slope Just like with means and proportions, confidence intervals can be made for slopes. These intervals are used to estimate the true value for the population slope.

Confidence Intervals for Slope Just like with hypothesis testing, the value for degrees of freedom is ___________ than the sample size. df = ______ Two fewer n – 2

Example: Age and Reading Distance The 95% confidence interval for the slope from the reading distance example is

Example: Age and Reading Distance The correct interpretation for this confidence interval is that we are 95% confident that the true ____________________for the linear relationship between reading distance and age is between _______ and ________. Does this agree with our conclusion from the hypothesis test? __________. population slope -3.88 -2.14 YES!

Correlation Remember, correlation (r) is a measure of _________ and _________ for a linear relationship As a note, if you find a significant hypothesis test for the population slope (so β1 ≠ 0), then the correlation will also be significantly different from zero. direction strength

Sample Size and Significance An important concept to keep in mind is that the larger the sample size, the more likely it is that significance would be found for a hypothesis test ___increases p-value decreases _________ decreases significance increases _____ increases significance increases n p-value n

If you understand today’s lecture… 14.1, 14.2, 14.4, 14.5, 14.7, 14.9, 14.21, 14.22, 14.24, 14.25, 14.27, 14.28 Objectives: • Formulate null and alternative hypotheses involving regression coefficients. • Calculate T-statistics; determine correct degrees of freedom.