Confidence Intervals: Slope of Regression Models - AP Statistics
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What is the margin of error for a $C%$ confidence interval for slope $\beta$?
What is the margin of error for a $C%$ confidence interval for slope $\beta$?
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$t^*,SE_b$. How far CI extends from point estimate.
$t^*,SE_b$. How far CI extends from point estimate.
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If a $C%$ CI for $\beta$ contains $0$, what conclusion should you make about a linear association?
If a $C%$ CI for $\beta$ contains $0$, what conclusion should you make about a linear association?
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Slope not statistically different from $0$ at level $\alpha$. Cannot reject null hypothesis of no linear relationship.
Slope not statistically different from $0$ at level $\alpha$. Cannot reject null hypothesis of no linear relationship.
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What is the correct interpretation template for a $C%$ CI for $\beta$ in context?
What is the correct interpretation template for a $C%$ CI for $\beta$ in context?
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$C%$ confident $\beta$ is between L and U (units of $y$ per $x$). States confidence that true slope falls within calculated bounds.
$C%$ confident $\beta$ is between L and U (units of $y$ per $x$). States confidence that true slope falls within calculated bounds.
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Which condition for a slope CI requires that observations are independent (often from random sampling)?
Which condition for a slope CI requires that observations are independent (often from random sampling)?
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Independence of observations. Ensures one observation doesn't influence another's error term.
Independence of observations. Ensures one observation doesn't influence another's error term.
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Which condition for a slope CI requires residuals to be approximately Normal for inference?
Which condition for a slope CI requires residuals to be approximately Normal for inference?
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Normality of residuals. Required for valid t-distribution inference about the slope.
Normality of residuals. Required for valid t-distribution inference about the slope.
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What is the general form of a $C%$ confidence interval for the population slope $\beta$?
What is the general form of a $C%$ confidence interval for the population slope $\beta$?
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$b \pm t^*,SE_b$. Uses sample slope plus/minus critical value times standard error.
$b \pm t^*,SE_b$. Uses sample slope plus/minus critical value times standard error.
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State the degrees of freedom used for a $t$ interval for the regression slope $\beta$.
State the degrees of freedom used for a $t$ interval for the regression slope $\beta$.
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$df=n-2$. Subtract 2 from sample size for slope inference.
$df=n-2$. Subtract 2 from sample size for slope inference.
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What does $s$ represent in the slope standard error formula $SE_b=\frac{s}{\sqrt{\sum (x_i-\bar{x})^2}}$?
What does $s$ represent in the slope standard error formula $SE_b=\frac{s}{\sqrt{\sum (x_i-\bar{x})^2}}$?
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$s=\sqrt{\frac{\sum e_i^2}{n-2}}$ (residual SD). Square root of sum of squared residuals divided by degrees of freedom.
$s=\sqrt{\frac{\sum e_i^2}{n-2}}$ (residual SD). Square root of sum of squared residuals divided by degrees of freedom.
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Identify the critical value symbol used in a $C%$ CI for slope $\beta$.
Identify the critical value symbol used in a $C%$ CI for slope $\beta$.
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$t^*=t_{\frac{\alpha}{2},,n-2}$. Critical value from t-distribution with $n-2$ df and area $\alpha/2$ in each tail.
$t^*=t_{\frac{\alpha}{2},,n-2}$. Critical value from t-distribution with $n-2$ df and area $\alpha/2$ in each tail.
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What is the parameter estimated by the slope confidence interval in simple linear regression?
What is the parameter estimated by the slope confidence interval in simple linear regression?
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$\beta$ (the true population slope). Greek letter beta represents the true slope in the population regression line.
$\beta$ (the true population slope). Greek letter beta represents the true slope in the population regression line.
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What is the meaning of $b$ in the interval $b \pm t^*SE_b$?
What is the meaning of $b$ in the interval $b \pm t^*SE_b$?
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$b$ is the least-squares sample slope. Calculated from sample data using least squares method.
$b$ is the least-squares sample slope. Calculated from sample data using least squares method.
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Which condition checks that the relationship between $x$ and $y$ is approximately straight for a slope CI?
Which condition checks that the relationship between $x$ and $y$ is approximately straight for a slope CI?
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Linearity (scatterplot shows linear trend). Ensures the model $y = \beta_0 + \beta x + \epsilon$ is appropriate.
Linearity (scatterplot shows linear trend). Ensures the model $y = \beta_0 + \beta x + \epsilon$ is appropriate.
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Which condition for a slope CI is assessed using a residual plot for constant spread?
Which condition for a slope CI is assessed using a residual plot for constant spread?
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Equal variance (homoscedasticity). Residuals should have roughly constant spread across all x values.
Equal variance (homoscedasticity). Residuals should have roughly constant spread across all x values.
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If a $C%$ CI for $\beta$ is entirely above $0$, what does that imply about the association?
If a $C%$ CI for $\beta$ is entirely above $0$, what does that imply about the association?
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Positive linear association; reject $H_0:\beta=0$. All plausible slopes are positive, indicating x and y increase together.
Positive linear association; reject $H_0:\beta=0$. All plausible slopes are positive, indicating x and y increase together.
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Identify the degrees of freedom for slope inference when $n=18$ paired observations are used.
Identify the degrees of freedom for slope inference when $n=18$ paired observations are used.
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$df=16$. Use $df = n - 2 = 18 - 2$ for regression slope inference.
$df=16$. Use $df = n - 2 = 18 - 2$ for regression slope inference.
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Choose the correct units for $\beta$ in a regression of $y$ (dollars) on $x$ (hours).
Choose the correct units for $\beta$ in a regression of $y$ (dollars) on $x$ (hours).
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dollars per hour. Slope units are always y-units divided by x-units.
dollars per hour. Slope units are always y-units divided by x-units.
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What data-collection condition is required to generalize a slope confidence interval to a population?
What data-collection condition is required to generalize a slope confidence interval to a population?
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Random sample or random assignment. Ensures results apply beyond the sample.
Random sample or random assignment. Ensures results apply beyond the sample.
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Compute the $95%$ CI for $\beta$ if $b=2.4$, $SE_b=0.5$, and $t^*=2.10$.
Compute the $95%$ CI for $\beta$ if $b=2.4$, $SE_b=0.5$, and $t^*=2.10$.
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$\left(1.35,\ 3.45\right)$. $2.4 \pm 2.10(0.5) = 2.4 \pm 1.05$.
$\left(1.35,\ 3.45\right)$. $2.4 \pm 2.10(0.5) = 2.4 \pm 1.05$.
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Find the margin of error if $t^*=2.00$ and $SE_b=0.30$ for a slope interval.
Find the margin of error if $t^*=2.00$ and $SE_b=0.30$ for a slope interval.
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$0.60$. $ME = 2.00 \times 0.30 = 0.60$.
$0.60$. $ME = 2.00 \times 0.30 = 0.60$.
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Identify whether $\beta=0$ is plausible if the $90%$ CI for $\beta$ is $(-0.8,\ 1.6)$.
Identify whether $\beta=0$ is plausible if the $90%$ CI for $\beta$ is $(-0.8,\ 1.6)$.
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Yes, because $0$ is in the interval. CI contains 0, so no linear relationship proven.
Yes, because $0$ is in the interval. CI contains 0, so no linear relationship proven.
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What is the point estimate for the population slope $\beta$ in a regression model?
What is the point estimate for the population slope $\beta$ in a regression model?
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$b$ (the least-squares slope). Sample statistic estimates population parameter.
$b$ (the least-squares slope). Sample statistic estimates population parameter.
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Identify the condition about the relationship between $x$ and $y$ needed for a slope confidence interval.
Identify the condition about the relationship between $x$ and $y$ needed for a slope confidence interval.
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Linearity of the mean response. Ensures the model form is appropriate.
Linearity of the mean response. Ensures the model form is appropriate.
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Identify the condition about the residuals’ spread needed for a slope confidence interval.
Identify the condition about the residuals’ spread needed for a slope confidence interval.
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Constant variance (homoscedasticity). Equal spread at all $x$ values.
Constant variance (homoscedasticity). Equal spread at all $x$ values.
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Identify the condition about residuals needed for a slope confidence interval when using $t$ procedures.
Identify the condition about residuals needed for a slope confidence interval when using $t$ procedures.
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Residuals are approximately Normal. Required for valid $t$ distribution inference.
Residuals are approximately Normal. Required for valid $t$ distribution inference.
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What parameter does a confidence interval for the slope in linear regression estimate?
What parameter does a confidence interval for the slope in linear regression estimate?
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$\beta$ (the true population slope). CIs estimate population parameters, not sample statistics.
$\beta$ (the true population slope). CIs estimate population parameters, not sample statistics.
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What degrees of freedom are used for a $t$ interval for the slope in simple linear regression?
What degrees of freedom are used for a $t$ interval for the slope in simple linear regression?
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$df=n-2$. Lose 2 df: one for intercept, one for slope.
$df=n-2$. Lose 2 df: one for intercept, one for slope.
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If a $C%$ CI for $\beta$ is entirely below $0$, what does that imply about the association?
If a $C%$ CI for $\beta$ is entirely below $0$, what does that imply about the association?
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Negative linear association; reject $H_0:\beta=0$. All plausible slopes are negative, indicating x increases as y decreases.
Negative linear association; reject $H_0:\beta=0$. All plausible slopes are negative, indicating x increases as y decreases.
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Compute the margin of error if $t^*=2.10$ and $SE_b=0.40$.
Compute the margin of error if $t^*=2.10$ and $SE_b=0.40$.
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$0.84$. Margin of error equals $t^* \times SE_b = 2.10 \times 0.40$.
$0.84$. Margin of error equals $t^* \times SE_b = 2.10 \times 0.40$.
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Find the $95%$ CI for $\beta$ if $b=3.2$, $t^*=2.00$, and $SE_b=0.50$.
Find the $95%$ CI for $\beta$ if $b=3.2$, $t^*=2.00$, and $SE_b=0.50$.
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$(2.2,,4.2)$. Calculate $3.2 \pm 2.00(0.50) = 3.2 \pm 1.0$.
$(2.2,,4.2)$. Calculate $3.2 \pm 2.00(0.50) = 3.2 \pm 1.0$.
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Identify whether there is evidence of a positive linear relationship if the CI for $\beta$ is $(0.2,\ 1.1)$.
Identify whether there is evidence of a positive linear relationship if the CI for $\beta$ is $(0.2,\ 1.1)$.
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Yes, because the entire interval is $>0$. All plausible values are positive.
Yes, because the entire interval is $>0$. All plausible values are positive.
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