Q: Given the following paired data for lottery jackpots (in millions) and tickets sold (in millions): $$x = [334, 127, 300, 227, 202, 180, 164, 145, 255]$$, $$y = [54, 16, 41, 27, 23, 18, 18, 16, 26]$$, the sample means are $$\overline{x} = 214.89$$ and $$\overline{y} = 26.56$$, and the sample standard deviations are $$s_x = 70.51$$ and $$s_y = 12.96. If $$the correlation coefficient is $$r = 0.947$$, what is the slope $$b_1 of $$the regression line?

$$b_1$$ = \dfrac{0.947 \times 12.96}{70.51} = 0.174$$

Q: Given the regression equation $$\hat{y} = -10.9 + 0.174x$$, what is the predicted number of tickets sold (in millions) when the jackpot is $$x = 625$$ million?

$$\hat{y} = -10.9 + 0.174 \times 625 = 97.9$$

Question 1

Which of the following statements best describes the meaning of the linear correlation coefficient $$r$$?

Accepted Answer

It measures the strength and direction of a linear relationship between two quantitative variables.

Question 2

Given the following paired data for lottery jackpots (in millions) and tickets sold (in millions): $$x = [334, 127, 300, 227, 202, 180, 164, 145, 255]$$, $$y = [54, 16, 41, 27, 23, 18, 18, 16, 26]$$, the sample means are $$\overline{x} = 214.89$$ and $$\overline{y} = 26.56$$, and the sample standard deviations are $$s_x = 70.51$$ and $$s_y = 12.96. If $$the correlation coefficient is $$r = 0.947$$, what is the slope $$b_1 of $$the regression line?

Accepted Answer

$$b_1$$ = \dfrac{0.947 	imes 12.96}{70.51} = 0.174$$

Question 3

Which of the following is NOT a requirement for using the sample linear correlation coefficient $$r to $$make inferences about the population?

Accepted Answer

The $$x$$ and $$y$$ variables must both be categorical.

Question 4

Suppose you calculate $$r = 0.947$$ for a sample of $$n = 9$$ pairs of data. Using a significance level of $$\alpha = 0.05$$ and a critical value of $$r = 0.666$$, what is the correct conclusion?

Accepted Answer

There is sufficient evidence to support the claim of a linear correlation.

Question 5

Which of the following best explains why 'correlation does not imply causation'?

Accepted Answer

A linear relationship between $$x$$ and $$y$$ can exist due to a lurking variable or coincidence, not necessarily because $$x$$ causes $$y.$$

Question 6

Given the regression equation $$\hat{y} = -10.9 + 0.174x$$, what is the predicted number of tickets sold (in millions) when the jackpot is $$x = 625$$ million?

Accepted Answer

$$\hat{y} = -10.9 + 0.174 	imes 625 = 97.9$$

Question 7

Which of the following is a property of the linear correlation coefficient $$r$$?

Accepted Answer

$$r is $$always between $$-1$$ and $$1$$ inclusive.

Question 8

A residual is defined as:

Accepted Answer

The difference between the observed $$y$$ value and the predicted $$y$$ value: $$y - \hat{y}.$$

Question 9

Which of the following scenarios would most likely result in a spurious correlation?

Accepted Answer

Two time-series variables that both increase over time but are unrelated.

Question 10

If the regression equation is $$\hat{y} = b_0$ + b_1x$$, what does the slope $$b_1$$ represent in a real-world context?

Accepted Answer

The marginal change in $$y$$ for a one-unit increase in $$x.$$

Question 11

Which of the following is NOT a correct interpretation of $$r^2 ($$the coefficient of determination)?

Accepted Answer

$$r^2$$ measures the probability that $$x$$ causes $$y.$$

Question 12

A residual plot is used to:

Accepted Answer

Assess whether the regression model is appropriate by checking for patterns in the residuals.

Question 13

Which of the following would indicate that a regression model is NOT appropriate based on the residual plot?

Accepted Answer

The residual plot shows a clear pattern or becomes much wider or thinner from left to right.

Question 14

Suppose you are testing the null hypothesis $$H_0$$: ho = 0$$ against the alternative H_1$: ho 
eq 0$$ for a sample of $$n = 15$$ pairs and you calculate $$r = 0.60. $$The test statistic is $$t = \dfrac{r\sqrt{n-2}}{\sqrt{1-r^2$}}. $$What is the value of the test statistic?

Accepted Answer

$$t = \dfrac{0.60 \sqrt{13}}{\sqrt{1-0.36}} = \dfrac{0.60 	imes 3.606}{\sqrt{0.64}} = \dfrac{2.164}{0.8} = 2.705$$

Question 15

Which of the following is a common error when interpreting correlation?

Accepted Answer

Assuming that correlation implies causality.

Question 16

If a single outlier is present in your data, what is the most likely effect on the value of $$r$$?

Accepted Answer

It can dramatically increase or decrease the value of $$r.$$

Question 17

Which formula is used to calculate the linear correlation coefficient $$r$$ for a sample of $$n$$ paired data?

Accepted Answer

$$r = \dfrac{n\sum xy - (\sum x)(\sum y)}{\sqrt{n\sum x^2$ - (\sum x)^2$} \sqrt{n\sum y^2$ - (\sum y)^2$}}$$

Question 18

In the context of regression, what is an influential point?

Accepted Answer

A data point that strongly affects the graph of the regression line.

Question 19

Which of the following is the best strategy for making predictions using a regression equation?

Accepted Answer

Use the regression equation only if the model fits the data well and the value of $$x is $$within the range of the sample data.

Chapter 10: Correlation and Regression – Study Notes

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