Making Inferences about Slopes Section 11.2 Making Inferences about Slopes

Making Inferences about Slopes Inference means . . . .

Making Inferences about Slopes Inference means using results from sample to make conclusion about the population

Linear Models μy = y = b0 + b1x “true” or theoretical regression line LSRL used to make predictions

Even when the true slope, , is 0, the estimate, b1, will usually _____________.

Even when the true slope, , is 0, the estimate, b1, will usually turn out to be different from 0.

Even when the true slope, , is 0, the estimate, b1, will usually turn out to be different from 0. A significance test for the slope of a regression line asks “Is that trend real, or could the numbers come out the way they did by chance?”

Test Statistic for Slope The test statistic for the slope, t, is the difference between the slope, b1, estimated from the sample, and the hypothesized slope, , measured in standard errors.

Test Statistic for Slope If a linear model is correct

Test Statistic for Slope If a linear model is correct and the null hypothesis is true, then

Test Statistic for Slope If a linear model is correct and the null hypothesis is true, then the test statistic has a t-distribution with n - 2 degrees of freedom.

Significance Test for a Slope Generally used when you have bivariate data from sample that appear to have a positive (or negative) linear association and you want to establish this association is “real”.

Significance Test for a Slope “Real” linear association is based on determining the nonzero slope you see did not happen just by chance. Why is this important?

Significance Test for a Slope Why is this important? A true linear relationship with a non-zero slope means knowing a value of x is helpful in predicting the value of y.

Components of Significance Test for a Slope How many components are in the test? What are they?

Components of Significance Test for a Slope How many components are in the test? 4 What are they?

Components of Significance Test for a Slope 4 components are in this test. 1) Name test and check conditions

Components of Significance Test for a Slope 4 components are in this test. 1) Name test and check conditions 2) State the hypotheses

Components of Significance Test for a Slope 4 components are in this test. 1) Name test and check conditions 2) State the hypotheses 3) Compute value of test statistic, find P-value, and draw sketch

Components of Significance Test for a Slope 4 components are in this test. 1) Name test and check conditions 2) State the hypotheses 3) Compute value of test statistic, find P-value, and draw sketch 4) Write conclusion in context linked to computations and in context

Name of test?

Name Test Two-sided significance test for a slope or One-sided significance test for a slope

Conditional distributions of y for fixed values of x

For test to work, the conditional distributions of y for fixed values of x must be approximately normal with means that lie near a line and standard deviations that are constant across all values of x Thus, we must check 4 conditions.

Conditions Randomness

Conditions Randomness Linearity

Conditions Randomness Linearity Uniform residuals

Conditions Randomness Linearity Uniform residuals Normality

First Condition Randomness: Verify you have one of these situations.

First Condition Randomness: Verify you have one of these situations. Single random sample from bivariate population -- situation we’ll see most often

First Condition Randomness: Verify you have one of these situations. i. Single random sample from bivariate population ii. A set of independent random samples, one for each fixed value of the explanatory variable, x

First Condition Randomness: Verify you have one of these situations. i. Single random sample from bivariate population ii. A set of independent random samples, one for each fixed value of the explanatory variable, x iii. Experiment with random assignment of treatments

Second Condition Linearity: Make a scatterplot and check to see if the relationship looks linear.

Second Condition Linearity: Make a scatterplot and check to see if the relationship looks linear. Note: On quiz or test, you must show the scatterplot with labels. Simply saying “based on scatterplot, relationship looks linear” gets no credit.

Third Condition Uniform residuals: Make a residual plot to check departures from linearity and that residuals are of uniform size across all values of x.

Third Condition Uniform residuals: Make a residual plot to check departures from linearity and that residuals are of uniform size across all values of x. Residual plot Scatterplot

Third Condition Uniform residuals: Make a residual plot to check departures from linearity and that residuals are of uniform size across all values of x. Note: On quiz or test, you must show the residual plot you analyzed or no credit.

Fourth Condition Normality: Make a univariate plot (dot plot, stemplot, boxplot or histogram) of the residuals to see if it’s reasonable to assume that the residuals came from a normal distribution.

Fourth Condition Normality: Make a univariate plot (dot plot, stemplot, boxplot or histogram) of the residuals to see if it’s reasonable to assume that the residuals came from a normal distribution.

Fourth Condition Normality: Make a univariate plot (dot plot, stemplot, boxplot or histogram) of the residuals to see if it’s reasonable to assume that the residuals came from a normal distribution. Note: On quiz or test, you must show the plot you analyzed with rationale - - no superficial statement

Step 2. State Hypotheses Null hypothesis will usually be: H0: = 0, where is slope of the true regression line. Note: It is possible the hypothesized value, , may be some constant other than 0.

Step 2. State Hypotheses Alternative hypothesis will usually be one of these three: Ha: 0, Ha: < 0, Ha: > 0

3. Compute Test Statistic, Find P-value, Draw Sketch

3. Compute Test Statistic, Find P-value, Draw Sketch

3. Compute Test Statistic, Find P-value, Draw Sketch

3. Compute Test Statistic, Find P-value, Draw Sketch

Slope of line = ?

Slope of line = 0

So, r = 0

: Greek letter “rho” r: English equivalent

: Greek letter “rho” r: English equivalent : b1 as : r

3. Compute Test Statistic, Find P-value, Draw Sketch

3. Compute Test Statistic, Find P-value, Draw Sketch

3. Compute Test Statistic, Find P-value, Draw Sketch Note that the calculator’s output does not include the standard error of the slope, .

3. Compute Test Statistic, Find P-value, Draw Sketch Note that the calculator’s output does not include the standard error of the slope, . However, because the null hypothesis is , the formula for the test statistic is

3. Compute Test Statistic, Find P-value, Draw Sketch Note that the calculator’s output does not include the standard error of the slope, . However, because the null hypothesis is , the formula for the test statistic is Therefore,

4. Write Conclusion The smaller the P-value, the stronger the evidence against the null hypothesis. Reject Ho if the P-value is less than the significance level of . Or compare the value of t to the critical value, t*. Reject Ho if ItI t*, for a two-sided test.

Page 766, P11

Page 766, P11 The scatterplot in Display 11.24 shows very weak correlation, so the slope will be close to 0. The t-value will be _______ (in absolute value) and the P-value will be ________.

Page 766, P11 The scatterplot in Display 11.24 shows very weak correlation, so the slope will be close to 0. The t-value will be small (in absolute value) and the P-value will be ________.

Page 766, P11 The scatterplot in Display 11.24 shows very weak correlation, so the slope will be close to 0. The t-value will be small (in absolute value) and the P-value will be large.

Page 766, P12 Mars rock data are on page 737 and soil data are on page 754.

Page 766, P12 Mars rock data are on page 737 and soil data are on page 754.

Page 766, P12

Page 766, P12 With 11 - 2 = 9 degrees of freedom, the P-value from the calculator is 0.00029.

Page 766, P12 With 11 - 2 = 9 degrees of freedom, the P-value from the calculator is 0.00029. 2[tcdf(5.6982, 1EE99, 9)]

Page 766, P12 With 11- 2 = 9 degrees of freedom, the P-value from the calculator is 0.00029. With a P-value this small, you reject the null hypothesis that the slope of the true linear relationship between percentage of sulfur and redness is zero.

Page 766, P13

Page 766, P13

Page 766, P13 temperature = 25.232 + 3.291 chirps/sec Linear model

Page 766, P13 a) temperature = 25.232 + 3.291 chirps/sec If the number of chirps per second increases by 1, we can expect the temperature to increase by about 3.291 oF.

Page 766, P13 b)

Page 766, P13 The residual plot shows no obvious pattern, so a linear model fits the data well. There is little evidence that the residuals tend to change in size as x increases.

Page 766, P13

Page 766, P13 The dot plot of the residuals shows no outliers or obvious skewness or any other indications of non-normality.

Page 766, P13 Have we checked all the conditions we should check?

Page 766, P13 Have we checked all the conditions we should check? No. Randomness: We can not tell if this was a random sample of cricket chirping.

Page 766, P13 c) Perform a two-sided LinRegTTest to determine t and a P-value. Start by stating the hypotheses.

Page 766, P13 c) Ho: = 0, there is no linear relationship between rate of chirping and temperature. Ha: 0

Page 766, P13 c) t = 5.47 P-value = 0.00011

Page 766, P13 c) t = 5.47; P-value = 0.00011

Page 766, P13 Conclusion?

Page 766, P13 I reject the null hypothesis because the P-value of 0.00011 is less than the significance level of α = 0.05.

Page 766, P13 I reject the null hypothesis because the P-value of 0.00011 is less than the significance level of α = 0.05. There is sufficient evidence to support the claim that there is a linear relationship between the rate of chirping and the air temperature. This conclusion depends on having a random sample of cricket chirping.

Questions?