Slope of a Regression Model (Test) - AP Statistics
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What is the typical two-sided alternative hypothesis for a test of slope in linear regression?
What is the typical two-sided alternative hypothesis for a test of slope in linear regression?
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$H_a: \beta \ne 0$. Tests for any linear relationship, positive or negative.
$H_a: \beta \ne 0$. Tests for any linear relationship, positive or negative.
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State the test statistic formula for testing the population slope in simple linear regression.
State the test statistic formula for testing the population slope in simple linear regression.
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$t = \frac{b-\beta_0}{SE_b}$. Standardizes difference between sample and hypothesized slope.
$t = \frac{b-\beta_0}{SE_b}$. Standardizes difference between sample and hypothesized slope.
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Which sampling distribution is used for the slope test statistic when conditions are met?
Which sampling distribution is used for the slope test statistic when conditions are met?
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$t$ distribution with $df = n-2$. Slope test statistic follows $t$ when conditions are met.
$t$ distribution with $df = n-2$. Slope test statistic follows $t$ when conditions are met.
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What is the condition on residuals that supports the Normality requirement for a slope test?
What is the condition on residuals that supports the Normality requirement for a slope test?
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Residuals are approximately Normal. Normal residuals support $t$ distribution assumption.
Residuals are approximately Normal. Normal residuals support $t$ distribution assumption.
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What is the decision rule using significance level $\alpha$ for a slope test?
What is the decision rule using significance level $\alpha$ for a slope test?
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Reject $H_0$ if $p\text{-value} < \alpha$. Standard hypothesis test decision rule.
Reject $H_0$ if $p\text{-value} < \alpha$. Standard hypothesis test decision rule.
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Identify the correct conclusion when $p=0.03$ and $\alpha=0.05$ for $H_a: \beta \ne 0$.
Identify the correct conclusion when $p=0.03$ and $\alpha=0.05$ for $H_a: \beta \ne 0$.
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Reject $H_0$; conclude evidence of nonzero slope. Since $0.03 < 0.05$, reject null hypothesis.
Reject $H_0$; conclude evidence of nonzero slope. Since $0.03 < 0.05$, reject null hypothesis.
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Find the degrees of freedom for a slope test when the sample size is $n=18$.
Find the degrees of freedom for a slope test when the sample size is $n=18$.
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$df = 16$. Apply formula: $df = 18-2 = 16$.
$df = 16$. Apply formula: $df = 18-2 = 16$.
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Compute the test statistic when $b=2.4$, $SE_b=0.6$, and $H_0: \beta=0$.
Compute the test statistic when $b=2.4$, $SE_b=0.6$, and $H_0: \beta=0$.
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$t = 4$. Apply formula: $t = \frac{2.4-0}{0.6} = 4$.
$t = 4$. Apply formula: $t = \frac{2.4-0}{0.6} = 4$.
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Identify the correct interpretation of $\beta$ in context for a regression slope test.
Identify the correct interpretation of $\beta$ in context for a regression slope test.
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Mean change in $y$ for a $1$-unit increase in $x$. Slope measures average $y$ change per unit $x$ change.
Mean change in $y$ for a $1$-unit increase in $x$. Slope measures average $y$ change per unit $x$ change.
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What data-collection condition supports independence for a regression slope test?
What data-collection condition supports independence for a regression slope test?
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Observations are independent (random sample/assignment). Independence ensures valid probability calculations.
Observations are independent (random sample/assignment). Independence ensures valid probability calculations.
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What residual plot feature supports the equal variance condition for a regression slope test?
What residual plot feature supports the equal variance condition for a regression slope test?
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Roughly constant spread of residuals across $x$. Equal variance ensures consistent standard error.
Roughly constant spread of residuals across $x$. Equal variance ensures consistent standard error.
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What scatterplot feature supports the linearity condition for a regression slope test?
What scatterplot feature supports the linearity condition for a regression slope test?
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Approximately linear relationship between $x$ and $y$. Linear model appropriate when pattern is straight.
Approximately linear relationship between $x$ and $y$. Linear model appropriate when pattern is straight.
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State the test statistic formula for testing the slope in simple linear regression.
State the test statistic formula for testing the slope in simple linear regression.
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$t = \frac{b - \beta_0}{SE_b}$. Standardizes the difference between sample slope and hypothesized value.
$t = \frac{b - \beta_0}{SE_b}$. Standardizes the difference between sample slope and hypothesized value.
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What alternative hypothesis matches the claim that $y$ decreases as $x$ increases?
What alternative hypothesis matches the claim that $y$ decreases as $x$ increases?
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$H_a: \beta < 0$. Negative slope means $y$ decreases with $x$.
$H_a: \beta < 0$. Negative slope means $y$ decreases with $x$.
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What does it mean, in context, to reject $H_0: \beta = 0$ in a slope test?
What does it mean, in context, to reject $H_0: \beta = 0$ in a slope test?
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Evidence of a linear relationship between $x$ and $y$. Rejecting means the data supports a non-zero slope.
Evidence of a linear relationship between $x$ and $y$. Rejecting means the data supports a non-zero slope.
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What plot is used to assess the linear and equal-variance conditions for regression inference?
What plot is used to assess the linear and equal-variance conditions for regression inference?
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Residual plot. Shows residuals vs. fitted values to check assumptions.
Residual plot. Shows residuals vs. fitted values to check assumptions.
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Identify the four inference conditions for a regression slope test in one-variable regression.
Identify the four inference conditions for a regression slope test in one-variable regression.
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Linear, Independent, Normal, Equal variance. LINE conditions ensure valid inference for regression.
Linear, Independent, Normal, Equal variance. LINE conditions ensure valid inference for regression.
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Which quantity is the standard error of the estimated slope used in the slope test?
Which quantity is the standard error of the estimated slope used in the slope test?
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$SE_b$. Measures variability in the slope estimate.
$SE_b$. Measures variability in the slope estimate.
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What are the degrees of freedom for the $t$ test of the slope in simple linear regression?
What are the degrees of freedom for the $t$ test of the slope in simple linear regression?
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$df = n - 2$. Lose 2 df: one for estimating slope, one for intercept.
$df = n - 2$. Lose 2 df: one for estimating slope, one for intercept.
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What sampling distribution is used for the slope test statistic under the null hypothesis?
What sampling distribution is used for the slope test statistic under the null hypothesis?
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$t$ distribution with $df = n - 2$. Test statistic follows $t$ when null is true and conditions are met.
$t$ distribution with $df = n - 2$. Test statistic follows $t$ when null is true and conditions are met.
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What alternative hypothesis matches the claim that $y$ increases as $x$ increases?
What alternative hypothesis matches the claim that $y$ increases as $x$ increases?
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$H_a: \beta > 0$. Positive slope means $y$ increases with $x$.
$H_a: \beta > 0$. Positive slope means $y$ increases with $x$.
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What is the typical alternative hypothesis for testing whether a linear relationship exists?
What is the typical alternative hypothesis for testing whether a linear relationship exists?
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$H_a: \beta \ne 0$. Two-sided test checks if slope differs from zero in either direction.
$H_a: \beta \ne 0$. Two-sided test checks if slope differs from zero in either direction.
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What are the parameter and null hypothesis for a test of regression slope?
What are the parameter and null hypothesis for a test of regression slope?
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Parameter: $\beta$; $H_0: \beta = 0$. Tests whether the true slope $\beta$ equals zero (no linear relationship).
Parameter: $\beta$; $H_0: \beta = 0$. Tests whether the true slope $\beta$ equals zero (no linear relationship).
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What value of $\beta_0$ is used in the test statistic when testing for no linear relationship?
What value of $\beta_0$ is used in the test statistic when testing for no linear relationship?
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$\beta_0 = 0$. Testing for no relationship means the hypothesized slope is zero.
$\beta_0 = 0$. Testing for no relationship means the hypothesized slope is zero.
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What conclusion wording is correct regarding causation after a significant slope test?
What conclusion wording is correct regarding causation after a significant slope test?
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Association only; do not claim causation. Regression shows association, not cause-and-effect.
Association only; do not claim causation. Regression shows association, not cause-and-effect.
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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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$t = 4$. $t = \frac{2.4 - 0}{0.6} = \frac{2.4}{0.6} = 4$.
$t = 4$. $t = \frac{2.4 - 0}{0.6} = \frac{2.4}{0.6} = 4$.
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Find the degrees of freedom for a slope test when the sample size is $n = 18$.
Find the degrees of freedom for a slope test when the sample size is $n = 18$.
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$df = 16$. $df = 18 - 2 = 16$ for simple linear regression.
$df = 16$. $df = 18 - 2 = 16$ for simple linear regression.
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Identify the correct decision at $\alpha = 0.05$ if the slope test $p$-value is $0.03$.
Identify the correct decision at $\alpha = 0.05$ if the slope test $p$-value is $0.03$.
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Reject $H_0$. $p = 0.03 < 0.05$, so reject at 5% significance level.
Reject $H_0$. $p = 0.03 < 0.05$, so reject at 5% significance level.
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What does it mean, in context, to fail to reject $H_0: \beta = 0$ in a slope test?
What does it mean, in context, to fail to reject $H_0: \beta = 0$ in a slope test?
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Insufficient evidence of a linear relationship. Cannot conclude a linear relationship exists.
Insufficient evidence of a linear relationship. Cannot conclude a linear relationship exists.
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Choose the correct two-sided $p$-value expression for a slope test given observed $t$ and $df$.
Choose the correct two-sided $p$-value expression for a slope test given observed $t$ and $df$.
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$p = 2P(T_{df} \ge |t|)$. Two-sided test doubles the one-tail probability beyond $|t|$.
$p = 2P(T_{df} \ge |t|)$. Two-sided test doubles the one-tail probability beyond $|t|$.
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