L23 & UNIT 4 REVIEW

Requirement	How to Check	What you hope to see
1.	Linear Relationship	Scatterplot	“Hot dog” shape
		Residual Plot	No pattern in the residuals
2.	Normal Error Term	Histogram of the Residuals	A shape that is approximately normal
3.	Constant Variance	Residual Plot	No megaphone shape in the residuals
4.	X$\mathrm{<br>}$’s are Known Constants	Cannot be checked directly	X$\mathrm{<br>}$’s should be measured accurately and precisely
5.	Observations are Independent	Cannot be checked directly	Knowing the value of one of the Y$\mathrm{<br><span\; face=""Helvetica\; Neue",\; Helvetica,\; Arial,\; sans-serif">’s</span>}$tells you nothing about any other points

 By the end of this lesson, you should be able to:

• 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.

REMEMBER:

*Residuals are "left-over" variation

*Residuals = observed y - predicted y

>the mean of all the residuals is always 0

>Linear Regression compares 2 quantitative variables

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$-1$ and 1$1$ 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$-1$ or 1$1$ indicates a stronger association than a value of r$�$ closer to zero.

Lesson 22 Recap

• In statistics, we write the linear regression equation as Y^=b0+b1X$\widehat{�}={�}_0+{�}_1�$ where b0${�}_0$ is the Y-intercept of the line and b1${�}_1$ is the slope of the line. The values of b0${�}_0$ and b1${�}_1$ 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 b0${�}_0$ and b1${�}_1$ 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=β0+β1X$�={�}_0+{�}_1�$ where β0${�}_0$ is the true y-intercept of the line and β1${�}_1$ 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−Y^=Y−(b0+b1X)$$��������=�-\widehat{�}=�-({�}_0+{�}_1�)$$

• 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 (β1${�}_1$) on whether or not it equals zero. The appropriate hypotheses for this test are:

  H0:Ha:β1=0β1≠0$$\begin{array}{cl}{�}_0:& {�}_1=0\\ {�}_{�}:& {�}_1\ne 0\end{array}$$

• 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.