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.

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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.

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**histogram 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.

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