Inference for Quantitative Data: Slopes · ⁨מסקנה עבור נתונים כמותיים: שיפועים⁩

Enduring Understanding (VAR-1): Given that variation may be random or not, conclusions are uncertain.

Learning Objective VAR-1.K: Identify questions suggested by variation in scatter plots. [Skill 1.A]

• VAR-1.K.1 Variation in points' positions relative to a theoretical line may be random or non-random.

Enduring Understanding (UNC-4): An interval of values should be used to estimate parameters, in order to account for uncertainty.

Learning Objective UNC-4.AC: Identify an appropriate confidence interval procedure for a slope of a regression model. [Skill 1.D]

• UNC-4.AC.1 Consider a response variable, $y$, that is linearly related to an explanatory variable, $x$. For a simple random sample of $n$ observations, the sample regression line, $\hat{y} = a + bx$, is an estimate of the population regression line $\mu_y = \alpha + \beta x$. For a particular observation, $(x_i, y_i)$, the residual from the sample regression line, $y_i - \hat{y}_i = y_i - (a + bx_i)$, is an estimate of $y_i - (\alpha + \beta x_i)$, the deviation of the response variable from the population regression line. For all points $(x, y)$ in the population, the standard deviation of all of the deviations of the response variable from the population regression line, $\sigma$, can be estimated by the standard deviation of the residuals from the sample regression line, $s = \sqrt{\dfrac{\sum\left(y_i - \hat{y}_i\right)^2}{n-2}}$. (Note: This formula uses $n-2$ in the denominator instead of $n-1$ because two parameters, $\alpha$ and $\beta$, must be estimated to obtain the predicted values from the least-squares regression line.)
• UNC-4.AC.2 For a simple random sample of $n$ observations, let $b$ represent the slope of a sample regression line. Then the mean of the sampling distribution for $b$ equals the population slope: $\mu_b = \beta$. The standard deviation of the sampling distribution for $b$ is $\sigma_b = \dfrac{\sigma}{\sigma_x \sqrt{n}}$, where $\sigma_x = \sqrt{\dfrac{\sum\left(x_i - \bar{x}\right)^2}{n}}$.
• UNC-4.AC.3 The appropriate confidence interval for the slope of a regression model is a $t$-interval for the slope.

Learning Objective UNC-4.AD: Verify the conditions to calculate confidence intervals for the slope of a regression model. [Skill 4.C]

• UNC-4.AD.1 In order to calculate a confidence interval to estimate the slope of a regression line, we must check the following:
  • a. The true relationship between $x$ and $y$ is linear. Analysis of residuals may be used to verify linearity.
  • b. The standard deviation for $y$, $\sigma_y$, does not vary with $x$. Analysis of residuals may be used to check for approximately equal standard deviations for all $x$.
  • c. To check for independence:
    • i. Data should be collected using a random sample or a randomized experiment.
    • ii. When sampling without replacement, check that $n \le 10\% N$.
  • d. For a particular value of $x$, the responses ($y$-values) are approximately normally distributed. Analysis of graphical representations of residuals may be used to check for normality.
    • i. If the observed distribution is skewed, $n$ should be greater than 30.

Learning Objective UNC-4.AE: Determine the given margin of error for the slope of a regression model. [Skill 3.D]

• UNC-4.AE.1 For the slope of a regression line, the margin of error is the critical value $\left(t^*\right)$ times the standard error ($SE$) of the slope.
• UNC-4.AE.2 The standard error for the slope of a regression line with sample standard deviation, $s$, is $SE = \dfrac{s}{s_x \sqrt{n-1}}$, where $s$ is the estimate of $\sigma$ and $s_x$ is the sample standard deviation of the $x$ values.

Learning Objective UNC-4.AF: Calculate an appropriate confidence interval for the slope of a regression model. [Skill 3.D]

• UNC-4.AF.1 The point estimate for the slope of a regression model is the slope of the line of best fit, $b$.
• UNC-4.AF.2 For the slope of a regression model, the interval estimate is $b \pm t^* \left(SE_b\right)$.

Enduring Understanding (UNC-4): An interval of values should be used to estimate parameters, in order to account for uncertainty.

Learning Objective UNC-4.AG: Interpret a confidence interval for the slope of a regression model. [Skill 4.B]

• UNC-4.AG.1 In repeated random sampling with the same sample size, approximately C% of confidence intervals created will capture the slope of the regression model, i.e., the true slope of the population regression model.
• UNC-4.AG.2 An interpretation for a confidence interval for the slope of a regression line should include a reference to the sample taken and details about the population it represents.

Learning Objective UNC-4.AH: Justify a claim based on a confidence interval for the slope of a regression model. [Skill 4.D]

• UNC-4.AH.1 A confidence interval for the slope of a regression model provides an interval of values that may provide sufficient evidence to support a particular claim in context.

Learning Objective UNC-4.AI: Identify the effects of sample size on the width of a confidence interval for the slope of a regression model. [Skill 4.A]

• UNC-4.AI.1 When all other things remain the same, the width of the confidence interval for the slope of a regression model tends to decrease as the sample size increases.

Enduring Understanding (VAR-7): The $t$-distribution may be used to model variation.

Learning Objective VAR-7.M: Calculate an appropriate test statistic for the slope of a regression model. [Skill 3.E]

• VAR-7.M.1 The distribution of the slope of a regression model assuming all conditions are satisfied and the null hypothesis is true (null distribution) is a $t$-distribution.
• VAR-7.M.2 For simple linear regression when random sampling from a population for the response that can be modeled with a normal distribution for each value of the explanatory variable, the sampling distribution of $t = \dfrac{b - \beta}{SE_b}$ has a $t$-distribution with degrees of freedom equal to $n - 2$. When testing the slope in a simple linear regression model with one parameter, the slope, the test for the slope has $df = n - 1$.

Enduring Understanding (DAT-3): Significance testing allows us to make decisions about hypotheses within a particular context.

Learning Objective DAT-3.M: Interpret the $p$-value of a significance test for the slope of a regression model. [Skill 4.B]

• DAT-3.M.1 An interpretation of the $p$-value of a significance test for the slope of a regression model should recognize that the $p$-value is computed by assuming that the null hypothesis is true, i.e., by assuming that the true population slope is equal to the particular value stated in the null hypothesis.

Learning Objective DAT-3.N: Justify a claim about the population based on the results of a significance test for the slope of a regression model. [Skill 4.E]

• DAT-3.N.1 A formal decision explicitly compares the $p$-value to the significance $\alpha$. If the $p$-value $\le \alpha$, then reject the null hypothesis, $H_0 : \beta = \beta_0$. If the $p$-value $> \alpha$, then fail to reject the null hypothesis.
• DAT-3.N.2 The results of a significance test for the slope of a regression model can serve as the statistical reasoning to support the answer to a research question about that sample.

This topic is intended to focus on the skill of selecting an appropriate inference procedure now that students have a range of options. Students should be given opportunities to practice when and how to apply all learning objectives relating to inference.