Which of the following best describes the relationship between correlation and slope of the regression line?

## Question 1: A statistics instructor wants to use the number of hours studied ## to predict exam scores in her class. She wants to use a linear regression ## model. Data from previous years shows that the correlation between these two ## variables is 0.76. Which of the following is the best response for whether or ## not the instructor should use linear regression to predict exam scores for a ## student who studied 10 hours for the final? # -> Linear regression could be appropriate if the scatterplot shows a clear linear relationship. # Yes, there is a high correlation, so it is appropriate to use linear regression. # No, because there is no way to prove that more hours of study causes higher exam scores. # Yes, because linear regression is the statistical method used to make predictions when you have bivariate quantitative data. ## Question 1: Of the four plots shown below, which one appears to show the ## weakest relationship between two variables? # -> (IV) # Relationship is weak because it is horizontal line!! and thus there are no direct relationship # (I) # WRONG -> (II) # (III) ## Question 2: The first plot below was created by plotting data collected on ## two variables. Then, the second plot was created using the same data but with ## different units for the dependent variable. In the context of linear ## regression, which of the following best describes the differences between the ## two plots? # The correlation coefficient for the second plot has a smaller absolute value, but the slopes of the linear relationships in the two plots are the same. # -> The slope of the linear relationship in the first plot has a larger absolute value, but the correlation coefficients for the two plots are the same. # The slope of the linear relationship in the first plot has a smaller absolute value, but the correlation coefficients for the two plots are the same. # Both the slopes and the regression coefficients are the same for the two plots. # The correlation coefficient for the second plot has a larger absolute value, but the slopes of the linear relationships in the two plots are the same. ## Question 2: Which of the following is false? # Correlation measures the strength of the linear association between two numerical variables. # -> If the correlation coefficient is 1, then the slope must be 1 as well. # Two numerical variables with a correlation of 0.01 have very weak linear association. # Correlation coefficient and the slope always have the same sign (positive or negative). ## Question 3: Which of the following is false? # -> The variability of residuals should increase as x increases. # A data point that has a negative residual is located below the regression line. # Residuals of linear models should be distributed nearly normally around 0. # The residuals plot (residuals vs. x) should show a random scatter around 0. ## Question 4: Sixteen student volunteers at Ohio State University drank a ## randomly assigned number beers. Thirty minutes later, a police officer ## measured their blood alcohol content (BAC) in grams of alcohol per deciliter ## of blood. The scatterplot displays the rela- tionship between BAC and number ## of beers consumed. Suppose a mistake was found in the data: the student who ## supposedly drank the highest number of beers (9 beers) actually only drank 6. ## His BAC was recorded correctly. In a new scatterplot, how would the strength ## of the association appear - compared to the strength of the association shown ## here? # Roughly the same as the strength of the association shown in the above scatterplot. # -> Weaker than the strength of the association shown in the above scatterplot. # Stronger than the strength of the association shown in the above scatterplot. # It’s impossible to tell. ## Question 4: An ambitious young student collected data on household ## electricity usage for a few families. After she plotted the data (shown ## below), the student observed that there did not appear to be a strong, ## positive linear relationship between the two variables as she had expected. ## The student still suspects that such a relationship exists - which of the ## following is the best advice an experienced statistician could give to the ## girl in order to help her investigate whether there is a linear relationship? # -> Collect electricity usage data for more families of sizes 1 through 5. # There is actually no practical strategy; whatever the strength of the association between these two variables, we cannot get a better idea of it just by collecting more data. # Plot the current data again, using a different scale for electricity usage. A poorly-chosen scale for this plot may be hiding a linear trend. # Collect one data point each for a family of size 6, 7, etc. in order to extend the plot off to the right. ## Question 5: The R2 for the linear regression of two variables x and y is ## 0.60. The variables are negatively associated. Which of the following the ## correct correlation coefficient? Choose the closest answer. -1 * sqrt(0.60) # -1 because of the negative association # -0.36 # 0.77 # 0.40 # 0.36 # -> -0.77 ## Question 6: The scatterplot on the right shows the relationship between ## percentage of white residents and percentage of households with a female head ## (where no husband is present) in all 50 US States and the District of ## Columbia (DC). Which of the below best describes the two points marked as DC ## and Hawaii? # -> Hawaii has higher leverage and is more influential than DC. # DC is more influential than Hawaii, but it has lower leverage than Hawaii. # DC and Hawaii should both be excluded from a simple linear regression analysis. # WRONG -> DC is not an outlier, and Hawaii is a leverage point. # WRONG -> Hawaii is not an outlier, and DC is not a leverage point. # Neither DC nor Hawaii appear to be leverage points. ## Question 7: A colleague needs some help with a statistics problem: He brings ## you the plot shown below, along with a correlation coefficient of 0.03 which ## he calculated himself. The plot shows two numerical variables which are ## obviously strongly related, and as a result your colleague is afraid he made ## a mistake calculating the correlation coefficient: that is, he was surprised ## to get an answer so close to 0. Given only this information, which of the ## following responses is the best to give your colleague? # -> The correlation coefficient measures the strength of the linear relationship, therefore two variables that have a strong non-linear association might still have a low correlation coefficient. # Your colleague must have made a mistake in his calculations. A much higher correlation coefficient is expected for variables that show a clear association. ## Question 8: Based on an observational study, a clinical psychologist finds ## that the relationship between the number of weeks spent in a therapy hospital ## and number of seizures per week is described by the following equation: ## Which of the following is the best interpretation of the slope estimate? # For each additional seizure per week, we would expect the average number of additional weeks spent in the therapy hospital to be higher by 0.91 seizures. # -> For each additional week spent in the therapy hospital, we would expect the average number of seizures per week to lower by 0.91 seizures. # Each additional week spent in a therapy hospital decreases the number of seizures per week by 0.91. # All patients start their treatment with at least 14.09 seizures per week. ## Question 9: The model below is for predicting the heart weight (in g) of cats ## from their body weight (in kg). The coefficients are estimated using a ## dataset of 144 domestic cats. The correlation between the heart and body ## weight is 0.8. Which of the following is false? # -> The slope estimate would not change if body weights were measured in pounds. # The intercept is meaningless in context of the data and only serves to adjust the height of the regression line. # The correlation coefficient would not change if body weights were measured in pounds. # The explanatory variable is body weight, and the response variable is heart weight. ## Question 10: The model below is for predicting the heart weight (in g) of ## cats from their gender (female and male). The coefficients are estimated ## using a dataset of 144 domestic cats. Which of the following is false? # -> The intercept is meaningless. # Gender is a significant predictor of heart weight in cats. # On average, male cats are expected to have hearts that weigh 2.12 grams more than female cats. # The expected heart weight for male cats is, on average, 11.32 grams. # If the regression equation is written y^=b0+b1x, then plugging in x=0 would give you the predicted heart weight for a female cat. ## Question 10: The model below is for predicting the heart weight (in g) of ## cats from their gender (female and male). The coefficients are estimated ## using a dataset of 144 domestic cats. Which of the following is false? # -> Because Pr(>|t|)=0 for the gender variable, we can say that gender is not a significant predictor of heart weight in cats. # Gender is a significant predictor of heart weight in cats. # The expected heart weight for female cats is 9.2 grams, on average. # On average, male cats are expected to have hearts that weigh 2.12 grams more than female cats. # The expected heart weight for male cats is, on average, 11.32 grams. ## Question 11: The linear model below is used for predicting poverty rate from ## high school graduation rate in the 51 states in the US (including DC). ## povertyˆ=64.68−0.62 HS grad rate ## High school graduation rate for North Carolina is 81.4% and the poverty rate ## is 13.1%. What is the residual for this observation? Choose the closest ## answer. ## A snippet of the data matrix is provided below, pay attention to the scale of ## the data in solving this question: HSgr <- 81.4 model.result <- 64.68 - 0.62 * HSgr 13.1 - model.result # -24.8 # 0 # 24.8 # 1.1 # -> -1.1 ## Question 11: Fill in the blanks: A data point that has a negative residual is ## located ________ the regression line. # above # -> below # on ## Question 12: We fit a linear regression model for predicting the best used ## price of 23 GMC pickup trucks from their list price, both measured in ## thousands. Which of the following is false based on this model output? # -> The 95% confidence interval for the slope can be calculated as 0.85±84.7×0.01. # TRUE For each additional $1,000 in the list price of a GMC pickup truck we would expect the best used price to be higher on average by $850. # TRUE The linear model is best_used_priceˆ=0.43+0.85 list_price. # TRUE List price is a significant predictor of the best used price. # TRUE The intercept is meaningless in this context. ## Question 12: Determine if I or II is higher, or if they are equal: ## The uncertainty associated with the slope estimate when ## I. there is a lot of scatter around the regression line ## II. there is very little scatter around the regression line # I and II are equal ## b1 +/- Tdf * SEb1 ## Standard error is higher the higher the scatterness of the data # II is higher # -> I is higher ## Question 13: Which of the following best describes SST (sum of squares total) ## in a regression? # Explained variability in the response variable. # Total variability in the explanatory variable. # Unexplained variability in the response variable. # -> Total variability in the response variable. # Strength of the model fit. ## Question 13: The following ANOVA output is for the linear model predicting ## nicotine content (in mg) from tar content (in mg). Which of the following is ## R2? Choose the closest answer. # R2 = explained variability / total variability 2869 / 3008 # 5% # 0.2% # -> 95% # 20% # 4%

What is the relationship between correlation and slope of regression line?

What is the relationship between simple linear regression and correlation?

What is the relationship between correlation and regression?

Is the correlation coefficient the same as the slope of the regression line?