Slope of a Regression Model (Setup) - AP Statistics
Card 1 of 30
Which condition addresses whether observations are independent for regression inference?
Which condition addresses whether observations are independent for regression inference?
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Independence of observations. Data collection method should ensure no systematic dependencies.
Independence of observations. Data collection method should ensure no systematic dependencies.
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Which condition checks that residuals are approximately normally distributed for inference?
Which condition checks that residuals are approximately normally distributed for inference?
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Normality of residuals. Required for valid $t$ distribution and p-values.
Normality of residuals. Required for valid $t$ distribution and p-values.
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Which condition checks that residuals have constant spread across $x$ values?
Which condition checks that residuals have constant spread across $x$ values?
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Equal variance (homoscedasticity). Residual plot should show consistent vertical spread.
Equal variance (homoscedasticity). Residual plot should show consistent vertical spread.
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Which condition checks that the relationship between $x$ and $y$ is approximately linear?
Which condition checks that the relationship between $x$ and $y$ is approximately linear?
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Linearity condition. Scatterplot should show a roughly linear pattern.
Linearity condition. Scatterplot should show a roughly linear pattern.
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What are the degrees of freedom for the $t$ test of a regression slope with sample size $n$?
What are the degrees of freedom for the $t$ test of a regression slope with sample size $n$?
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$df = n-2$. Lose 2 degrees of freedom for estimating intercept and slope.
$df = n-2$. Lose 2 degrees of freedom for estimating intercept and slope.
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What distribution is used for the slope test statistic when conditions are met?
What distribution is used for the slope test statistic when conditions are met?
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A $t$ distribution. The standardized slope follows $t$ when regression conditions hold.
A $t$ distribution. The standardized slope follows $t$ when regression conditions hold.
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What value of $\beta_0$ is used in the test statistic for the common null $H_0: \beta=0$?
What value of $\beta_0$ is used in the test statistic for the common null $H_0: \beta=0$?
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$\beta_0 = 0$. Under the null hypothesis, the population slope equals zero.
$\beta_0 = 0$. Under the null hypothesis, the population slope equals zero.
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What is the sample statistic used to estimate the slope parameter $\beta$?
What is the sample statistic used to estimate the slope parameter $\beta$?
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The least-squares slope $b$. Sample slope $b$ estimates the population slope $\beta$.
The least-squares slope $b$. Sample slope $b$ estimates the population slope $\beta$.
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What alternative hypothesis matches testing for a negative linear association?
What alternative hypothesis matches testing for a negative linear association?
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$H_a: \beta < 0$. One-sided test for when $y$ decreases as $x$ increases.
$H_a: \beta < 0$. One-sided test for when $y$ decreases as $x$ increases.
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What alternative hypothesis matches testing for a positive linear association?
What alternative hypothesis matches testing for a positive linear association?
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$H_a: \beta > 0$. One-sided test for when $y$ increases as $x$ increases.
$H_a: \beta > 0$. One-sided test for when $y$ increases as $x$ increases.
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What alternative hypothesis matches a two-sided slope test for linear association?
What alternative hypothesis matches a two-sided slope test for linear association?
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$H_a: \beta \ne 0$. Two-sided test checks for any linear relationship, positive or negative.
$H_a: \beta \ne 0$. Two-sided test checks for any linear relationship, positive or negative.
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What null hypothesis is most common for a test of a regression slope?
What null hypothesis is most common for a test of a regression slope?
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$H_0: \beta = 0$. Tests whether there's no linear relationship between variables.
$H_0: \beta = 0$. Tests whether there's no linear relationship between variables.
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What is the parameter tested when you test the slope in a linear regression model?
What is the parameter tested when you test the slope in a linear regression model?
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The population slope $\beta$. We test the true population slope, not the sample slope.
The population slope $\beta$. We test the true population slope, not the sample slope.
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Find and correct the parameter error: Using $H_0: b=0$ to test for linear association.
Find and correct the parameter error: Using $H_0: b=0$ to test for linear association.
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Correct: $H_0: \beta = 0$. Must test population parameter $\beta$, not sample statistic $b$.
Correct: $H_0: \beta = 0$. Must test population parameter $\beta$, not sample statistic $b$.
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Compute the test statistic if $b=2.4$, $SE_b=0.6$, and $H_0: \beta=0$.
Compute the test statistic if $b=2.4$, $SE_b=0.6$, and $H_0: \beta=0$.
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$t = 4$. Calculate $t = \frac{2.4-0}{0.6} = 4$.
$t = 4$. Calculate $t = \frac{2.4-0}{0.6} = 4$.
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What are the degrees of freedom for a slope test when $n=18$ data pairs are used?
What are the degrees of freedom for a slope test when $n=18$ data pairs are used?
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$df = 16$. Calculate as $n-2 = 18-2 = 16$.
$df = 16$. Calculate as $n-2 = 18-2 = 16$.
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Identify the correct hypotheses for testing any linear association between $x$ and $y$ via slope.
Identify the correct hypotheses for testing any linear association between $x$ and $y$ via slope.
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$H_0: \beta = 0;\ H_a: \beta \ne 0$. Tests if slope differs from zero in either direction.
$H_0: \beta = 0;\ H_a: \beta \ne 0$. Tests if slope differs from zero in either direction.
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Identify the correct hypotheses for testing a positive association between $x$ and $y$ via slope.
Identify the correct hypotheses for testing a positive association between $x$ and $y$ via slope.
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$H_0: \beta = 0;\ H_a: \beta > 0$. Tests if slope is positive (right-tailed test).
$H_0: \beta = 0;\ H_a: \beta > 0$. Tests if slope is positive (right-tailed test).
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What is the test statistic form for testing a regression slope?
What is the test statistic form for testing a regression slope?
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$t = \frac{b-\beta_0}{SE_b}$. Standardizes the difference between sample and hypothesized slopes.
$t = \frac{b-\beta_0}{SE_b}$. Standardizes the difference between sample and hypothesized slopes.
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What parameter is tested when you test the slope in a linear regression model?
What parameter is tested when you test the slope in a linear regression model?
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The population slope $\beta$. The sample slope $b$ estimates the population slope $\beta$.
The population slope $\beta$. The sample slope $b$ estimates the population slope $\beta$.
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Identify the correct response variable and explanatory variable symbols in regression.
Identify the correct response variable and explanatory variable symbols in regression.
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Response $y$, explanatory $x$. $y$ is predicted by $x$ in regression notation.
Response $y$, explanatory $x$. $y$ is predicted by $x$ in regression notation.
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Which sampling distribution is used for the slope test statistic when assumptions hold?
Which sampling distribution is used for the slope test statistic when assumptions hold?
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A $t$ distribution with $df = n - 2$. The $t$ distribution accounts for estimating variance from the sample.
A $t$ distribution with $df = n - 2$. The $t$ distribution accounts for estimating variance from the sample.
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What are the degrees of freedom for the $t$ test of a regression slope with sample size $n$?
What are the degrees of freedom for the $t$ test of a regression slope with sample size $n$?
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$df = n - 2$. Loses 2 degrees: one for slope, one for intercept.
$df = n - 2$. Loses 2 degrees: one for slope, one for intercept.
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In the common slope test, what value is used for $\beta_0$ in $t = \frac{b-\beta_0}{SE_b}$?
In the common slope test, what value is used for $\beta_0$ in $t = \frac{b-\beta_0}{SE_b}$?
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$\beta_0 = 0$. Testing for no linear relationship means $\beta_0 = 0$.
$\beta_0 = 0$. Testing for no linear relationship means $\beta_0 = 0$.
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State the test statistic formula for testing a regression slope using sample slope $b$.
State the test statistic formula for testing a regression slope using sample slope $b$.
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$t = \frac{b - \beta_0}{SE_b}$. Standardizes the difference between sample and hypothesized slope.
$t = \frac{b - \beta_0}{SE_b}$. Standardizes the difference between sample and hypothesized slope.
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What is the usual null hypothesis for a test of slope in linear regression?
What is the usual null hypothesis for a test of slope in linear regression?
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$H_0: \beta = 0$. Tests whether there's no linear relationship between variables.
$H_0: \beta = 0$. Tests whether there's no linear relationship between variables.
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What alternative hypothesis matches the claim "there is a linear relationship"?
What alternative hypothesis matches the claim "there is a linear relationship"?
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$H_a: \beta \ne 0$. Two-sided test for any linear relationship, positive or negative.
$H_a: \beta \ne 0$. Two-sided test for any linear relationship, positive or negative.
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What alternative hypothesis matches the claim "as $x$ increases, $y$ tends to increase"?
What alternative hypothesis matches the claim "as $x$ increases, $y$ tends to increase"?
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$H_a: \beta > 0$. One-sided test for a positive linear relationship.
$H_a: \beta > 0$. One-sided test for a positive linear relationship.
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What alternative hypothesis matches the claim "as $x$ increases, $y$ tends to decrease"?
What alternative hypothesis matches the claim "as $x$ increases, $y$ tends to decrease"?
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$H_a: \beta < 0$. One-sided test for a negative linear relationship.
$H_a: \beta < 0$. One-sided test for a negative linear relationship.
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What is the correct wording of the parameter in context for a slope test?
What is the correct wording of the parameter in context for a slope test?
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$\beta$ is the true change in mean $y$ per $1$ unit increase in $x$. Interprets slope as the rate of change in the mean response.
$\beta$ is the true change in mean $y$ per $1$ unit increase in $x$. Interprets slope as the rate of change in the mean response.
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