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Making Inferences about Slopes
Section 11.2 Making Inferences about Slopes
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Making Inferences about Slopes
Inference means
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Making Inferences about Slopes
Inference means using results from sample to make conclusion about the population
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Linear Models μy = y = b0 + b1x “true” or theoretical regression line
LSRL used to make predictions
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Even when the true slope, , is 0, the estimate, b1, will usually _____________.
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Even when the true slope, , is 0, the estimate, b1, will usually turn out to be different from 0.
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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?”
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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.
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Test Statistic for Slope
If a linear model is correct
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Test Statistic for Slope
If a linear model is correct and the null hypothesis is true, then
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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.
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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”.
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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?
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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.
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Components of Significance Test for a Slope
How many components are in the test? What are they?
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Components of Significance Test for a Slope
How many components are in the test? 4 What are they?
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Components of Significance Test for a Slope
4 components are in this test. 1) Name test and check conditions
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Components of Significance Test for a Slope
4 components are in this test. 1) Name test and check conditions 2) State the hypotheses
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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
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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
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Name of test?
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Name Test Two-sided significance test for a slope or
One-sided significance test for a slope
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Conditional distributions of y for fixed values of x
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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.
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Conditions Randomness
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Conditions Randomness Linearity
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Conditions Randomness Linearity Uniform residuals
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Conditions Randomness Linearity Uniform residuals Normality
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First Condition Randomness: Verify you have one of these situations.
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First Condition Randomness: Verify you have one of these situations.
Single random sample from bivariate population -- situation we’ll see most often
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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
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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
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Second Condition Linearity: Make a scatterplot and check to see if the relationship looks linear.
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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.
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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.
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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
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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.
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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.
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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.
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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
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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.
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Step 2. State Hypotheses Alternative hypothesis will usually be one of these three: Ha: , Ha: < 0, Ha: > 0
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3. Compute Test Statistic, Find P-value, Draw Sketch
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3. Compute Test Statistic, Find P-value, Draw Sketch
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3. Compute Test Statistic, Find P-value, Draw Sketch
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3. Compute Test Statistic, Find P-value, Draw Sketch
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Slope of line = ?
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Slope of line = 0
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So, r = 0
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: Greek letter “rho” r: English equivalent
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: Greek letter “rho” r: English equivalent : b1 as : r
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3. Compute Test Statistic, Find P-value, Draw Sketch
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3. Compute Test Statistic, Find P-value, Draw Sketch
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3. Compute Test Statistic, Find P-value, Draw Sketch
Note that the calculator’s output does not include the standard error of the slope,
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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
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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,
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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.
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Page 766, P11
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Page 766, P11 The scatterplot in Display 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 ________.
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Page 766, P11 The scatterplot in Display 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 ________.
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Page 766, P11 The scatterplot in Display 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.
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Page 766, P12 Mars rock data are on page 737 and soil data are on page 754.
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Page 766, P12 Mars rock data are on page 737 and soil data are on page 754.
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Page 766, P12
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Page 766, P12 With 11 - 2 = 9 degrees of freedom,
the P-value from the calculator is
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Page 766, P12 With 11 - 2 = 9 degrees of freedom,
the P-value from the calculator is 2[tcdf(5.6982, 1EE99, 9)]
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Page 766, P12 With 11- 2 = 9 degrees of freedom,
the P-value from the calculator is 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.
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Page 766, P13
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Page 766, P13
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Page 766, P13 temperature = chirps/sec Linear model
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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 oF.
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Page 766, P13 b)
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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.
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Page 766, P13
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Page 766, P13 The dot plot of the residuals shows
no outliers or obvious skewness or any other indications of non-normality.
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Page 766, P13 Have we checked all the conditions we should check?
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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.
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Page 766, P13 c) Perform a two-sided LinRegTTest to determine t and a P-value. Start by stating the hypotheses.
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Page 766, P13 c) Ho: = 0, there is no linear relationship between rate of chirping and temperature. Ha:
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Page 766, P13 c) t = 5.47 P-value =
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Page 766, P13 c) t = 5.47; P-value =
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Page 766, P13 Conclusion?
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Page 766, P13 I reject the null hypothesis because the
P-value of is less than the significance level of α = 0.05.
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Page 766, P13 I reject the null hypothesis because the
P-value of 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.
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Questions?
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