Lesson 24: Review for Exam 4

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You are strongly encouraged to review the readings, homework exercises, and other activities from Units 1-4 as you prepare for the exam. In particular, you should go over the Review for Exam 1, the Review for Exam 2, and the Review for Exam 3. Use the Index to review definitions of important terms.

1 Lesson Outcomes

The expectation on the exam is the following outcomes. You should be able to do:

  • All of the Outcomes from Lesson 08 (Unit 1), Lesson 15 (Unit 2), and Lesson 20 (Unit 3)
  • Create a scatterplot of bivariate data.
  • Interpret the overall pattern in a scatter plot to assess linearity and direction.
  • Calculate the correlation coefficient.
  • Interpret the correlation coefficient, $r$, as a measure of strength and direction of a linear relationship between two variables.
  • Identify the explanatory and response variable in a study.
  • Calculate the slope and intercept of a regression model.
  • Interpret the slope of the regression model.
  • Make predictions using a regression model
  • Confidence Intervals for the slope of the regression line:
    • Calculate and interpret a confidence interval for the slope of the regression line given a confidence level.
    • Identify a point estimate and margin of error for the confidence interval.
    • Show the appropriate connections between the numerical and graphical summaries that support the confidence interval.
    • Check the requirements for the confidence interval.
  • Hypothesis Testing for the slope of the regression line:
    • State the null and alternative hypothesis.
    • Calculate the test-statistic, degrees of freedom and p-value of the hypothesis test.
    • Assess the statistical significance by comparing the p-value to the α-level.
    • Check the requirements for the hypothesis test.
    • Show the appropriate connections between the numerical and graphical summaries that support the hypothesis test.
    • Draw a correct conclusion for the hypothesis test.

2 Lesson Summaries

Click on the link at right for a review of the summaries from each lesson.

Here are the summaries for each lesson in unit 4. Reviewing these key points from each lesson will help you in your preparation for the exam.

Lesson 21 Recap
  • Creating scatterplots of bivariate data allows us to visualize the data by helping us understand its shape (linear or nonlinear), direction (positive, negative, or neither), and strength (strong, moderate, or weak).
  • The correlation coefficient ($r$) is a number between $-1$ and $1$ that tells us the direction and strength of the linear association between two variables. A positive $r$ corresponds to a positive association while a negative $r$ corresponds to a negative association. A value of $r$ closer to $-1$ or $1$ indicates a stronger association than a value of $r$ closer to zero.
  • The covariance is a measure of how two variables vary together. The formula for the covariance is $s_{xy}=r \cdot s_x \cdot s_y$.

Lesson 22 Recap
  • In statistics, we write the linear regression equation as $\hat Y=b_0+b_1X$ where $b_0$ is the Y-intercept of the line and $b_1$ is the slope of the line. The values of $b_0$ and $b_1$ are calculated using software.
  • Linear regression allows us to predict values of $Y$ for a given $X$. This is done by first calculating the coefficients $b_0$ and $b_1$ and then plugging in the desired value of $X$ and solving for $Y$.
  • The independent (or explanatory) variable ($X$) is the variable which is not affected by what happens to the other variable. The dependent (or response) variable ($Y$) is the variable which is affected by what happens to the other variable. For example, in the correlation between number of powerboats and number of manatee deaths, the number of deaths is affected by the number of powerboats in the water, but not the other way around. So, we would assign $X$ to represent the number of powerboats and $Y$ to represent the number of manatee deaths.

Lesson 23 Recap
  • The unknown true linear regression line is $Y=\beta_0+\beta_1X$ where $\beta_0$ is the true y-intercept of the line and $\beta_1$ is the true slope of the line.
  • A residual is the difference between the observed value of $Y$ for a given $X$ and the predicted value of $Y$ on the regression line for the same $X$. It can be expressed as:

$$ Residual = Y - \hat Y = Y - (b_0 + b_1 X) $$

  • To check all the requirements for bivariate inference you will need to create a scatterplot of $X$ and $Y$, a residual plot, and a Q-Q plot of the residuals.
  • We conduct a hypothesis test on bivariate data to know if there is a linear relationship between the two variables. To determine this, we test the slope ($\beta_1$) on whether or not it equals zero. The appropriate hypotheses for this test are:

$$ \begin{array}{1cl} H_0: & \beta_1=0 \\ H_a: & \beta_1\ne0 \end{array} $$

  • For bivariate inference we use software to calculate the sample coefficients, residuals, test statistic, $P$-value, and confidence intervals of the true linear regression coefficients.

3 Navigation

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Inference for Bivariate Data
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Lesson 24:
Review for Exam 4