Psych 10 / Stats 60, Practice Problem Set 10 (Week 10 Material), Solutions

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1 Psych 10 / Stats 60, Practice Problem Set 10 (Week 10 Material), Solutions Part 1: Conceptual ideas about correlation and regression Tintle The association would be negative (as distance increases, final exam score decreases) I would expect a positive relationship, such that as temperature increases, ice cream sales increase too. (Although you also might hypothesize that if temperatures get high enough then people might stay indoors and sales would actually decrease, or you might hypothesize that your range of temperature is so restricted that you actually wouldn t see a strong relationship within this range of summer temperatures) The correlation would be It is positive because students with above-average exam scores on the first exam will also be the students with above-average exam scores on the second exam. The magnitude is 1.00 because we could draw a straight line that goes through every single point, meaning there is a perfect linear relationship (described by the regression line y = * x, note the positive slope of 1) The relationship is not linear, and correlation only measures the linear relationship between two variables. The best fit line through this curve does not capture the data very well D (there could still be a relationship, just not a linear relationship) A (in general, although this won t be the case for every single observation) a. There is a strong and negative relationship between GDP per capita and infant mortality, but the relationship is not linear. Specifically, infant mortality drops off very steeply as GDP per capita increases from about 0 to 10 (thousand dollars), and then drops off very shallowly as GDP per capita increases beyond that. b. The regression line is a straight line and the relationship between the two variables is not linear, so the regression line will not give as good of a prediction as a line that was allowed to be curved. (Additionally, it will systematically under-predict mortality at low GDP, systematically over-predict mortality for middle GDP, and then systematically under-predict mortality / stop making sense at high GDP (infant mortality cannot be negative), which is not a good quality of a regression line we want the errors (residuals) to be similarly distributed at any point in our X variable). c. There is a strong and negative relationship between log(gdp per capita) and log(infant mortality), and now the relationship looks linear (follows a straight line). d. Yes, because the regression line is a straight line, the relationship between the two variables is linear, and the relationship is strong enough that we can make a good prediction for Y (log of infant mortality) based on X (log of GDP per capita). 1. The scatter plots below visualize sets of observations of (X,Y) pairs that have correlation coefficients of approximately r = -.50, r = 0, r = +.50, and r = +.80.

2 Which panel has which correlation? (For additional practice with this, I recommend using the applet found here: Panel A is.80 (the relationship is positive, and we can see it is the strongest of the relationships pictured here). Panel C is 0 (looks like a circle of points, suggesting no relationship), and then we can figure out that Panel B is -.50 and Panel D is +.50, since the relationship in Panel B is negative and the relationship in Panel D is positive. They also look weaker than Panel A but stronger than Panel C, suggesting they correspond to r = Tintle : Two researchers want to investigate whether there is a relationship between annual company profit (in dollars) and median annual salary paid by the company (in dollars). Researcher Bart collects data on a random sample of 40 companies, and researcher Lisa collects data on a random sample of 140 companies. After analyzing their respective data sets, each finds a correlation coefficient of r = If they each calculate a p-value corresponding to the probability of finding a correlation coefficient as or more extreme as 0.60 (if the population correlation coefficient, ρ, is 0), who will have a smaller p-value and why? a. Bart b. Lisa c. Both will have the same p-value d. More information is needed to answer this question The t-statistic (and thus the p-value) increases with r and with n. Since r is the same for both samples but n is larger for Lisa s sample (140 vs. 40), we know that her t- c d a b X Y

3 statistic will be larger and her p-value will be smaller. (It s less likely to find a correlation coefficient as or more extreme as r =.60 by random chance in a sample of 140 people (with ρ = 0) vs. a sample of 40 people (with ρ = 0)). 3. How (if it all) will a t-statistic corresponding to a correlation coefficient (r) change with: a. the sample size (n) t will increase as n increases b. the correlation coefficient (r) t will increase as r increases (and t will be positive if r is positive, and negative if r is negative) c. a t-statistic corresponding to the slope of the regression line (b) for the same data these t-statistics are identical d. a t-statistic corresponding to the intercept (a) of the regression line for the data a t-statistic for the intercept would be testing whether the expected value for y when x = 0 is significantly different from some hypothesized value (usually 0). the intercept doesn t automatically give us information about the slope or the correlation coefficient, and so we can t predict how a change in the t-statistic corresponding to the intercept would map to a change in the t-statistic corresponding to the slope or the correlation coefficient 4. What s a residual, both in words and using a formula? How is the best fitting regression line selected (what makes a line the best fitting)? A residual is the difference between an actual value of y that was observed and the predicted value of y from a regression line (ŷ predicted from the corresponding x), i.e., a residual is (y i - ŷ i ). The best fitting line is the line that minimizes the sum of the squared residuals when summing over all of the observations in the dataset, Ʃ i (y i - ŷ i ) 2, also called SS unexplained or SS error or SS residual (analogous to SS within in an ANOVA). 5. In words, describe how to interpret the intercept and slope of a regression line. The intercept is the expected value of y (our best guess for y, or the mean of y) when x = 0. The slope is how much of a change in y we would expect when x increases by In baseball, the correlation between players batting averages between two consecutive years has been estimated as r =.41. If a player has a batting average that is two standard deviations above the mean in 2016, what is our best guess for how many standard deviations above or below the mean the player s batting average will be in 2017? We can use ẑ y = r * z x. Since the player had a batting average that was two standard deviations above the mean, z x = +2.00, and thus ẑ y =.41 * 2 = 0.82, or.82 standard deviations above the mean. (In other words, we would predict a substantial decrease in the player s relative standing in the league). 7. If the correlation between schools performances on a state-wide test between two consecutive years is r =.80, and a school performs one standard deviation below the mean in 2016, what is our best guess for how many standard deviations above

4 or below the mean the school will perform in 2017? How would this change if the correlation was r =.50? What if the correlation was r = 0? And finally, what if the correlation was r = 1? We can use ẑ y = r * z x. Since the school was one standard deviation below the mean, z x = -1.00, and ẑ y =.80 * (-1) = -.80, or.80 standard deviations below the mean. If r =.50, this prediction would change to ẑ y =.50 * (-1) = If r = 0, this prediction would change to ẑ y = 0 * (-1) = 0. If r = 1, this prediction would change to ẑ y =.1 * (-1) = -1. (In other words, unless r = 1, we would expect an increase in the school s relative standing among other schools). 8. If the correlation between two variables is 0, what is our best guess for the value of y (ŷ) for any observation? Our best guess is the mean of y (ȳ). This is easiest to see if we think about regression with z-scores, ẑ y = r * z x = 0 * z x = 0, regardless of the value of z x. As a reminder, a z-score of 0 corresponds to the mean of the observations, so z y = 0 corresponds to ȳ. 9. What general principle explains the answers to the last three questions and how? These examples all illustrate regression to the mean. If there is no relationship between two variables then our best guess for any y is ȳ, regardless of how many standard deviations above or below the mean x is (ẑ y = 0). If there is a perfect relationship between two variables then we expect a pair of y and x from the same observation to be the same number of standard deviations above or below their respective means (ẑ y = z x ; note that this is our best guess, not that we would see this for all observations). And if the correlation coefficient is between 0 and 1, our best guess for y moves closer toward the mean of y (and away from a prediction based on z x ) as the relationship between the two variables gets weaker (ẑ y = r * z x ). 10. Tintle : For a given data set, a test of association based on a slope is equivalent to a test of association based on a correlation coefficient. Being equivalent means which of the following is true? a. the confidence intervals for the population correlation and population slope will be the same b. the observed correlation will be the same as the observed slope of the regression line c. the p-value will be the same whether you use correlation as the statistic or the slope of the regression line as the statistic d. all of the above The confidence intervals are in units of r and units of b, so it would not make sense for them to be equal. The observed correlation is only guaranteed to be the same as the observed slope of the regression line if all x and y have been converted to z- scores (z x and z y ). When working with real-world units it would be very reasonable to have a slope such that b > 1, but it is impossible for r > 1.

5 Part 2: Correlation and regression by hand Calculate the correlation coefficient (r) and the best fitting regression line for the set of data below. To help, x = 3, s x = 2.74, ȳ = 4, and s y = 1.58: X Y (x x ) (y - ȳ) (x x )(y - ȳ) = = -2 (-3)(-2) = = = 0 (-2)(0) = = = 1 (1)(1) = = = -1 (0)(-1) = = = 2 (4)(2) = 8 r = (1 / (n-1)) * (Ʃ(x - x )(y - ȳ))/ (s x * s y ) = (1 / (5-1)) * (15) / (2.74 * 1.58) = (1/4) * 15 / (2.74 * 1.58) =.87 (note, there are several formulas that can be used to calculate r, see lecture slides for additional options) b = r * (s y / s x ) =.87 * (1.58 / 2.74) =.50 a = ȳ - bx = 4 (.50)*(3) = 2.5 regression line is ŷ = x Calculate the correlation coefficient (r) and the best fitting regression line for the set of data below. Then, calculate the predicted value (ŷ) and the residual for each observed value of Y. To help, x = 2, s x = 0.816, ȳ = 3, and s y = 1.63: X Y (x x ) (y - ȳ) (x x )(y - ȳ) ŷ residual (y-ŷ) = = 0 (-1)(0) = *1 = = = = 2 (1)(2) = *3 = = = = -2 (0)(-2) = *2 = = = = 0 (0)(0) = *2 = = 0 r = (1 / (n-1)) * (Ʃ(x - x )(y - ȳ))/ (s x * s y ) = (1 / (4-1)) * (2) / (0.816 * 1.63) = (1/3) * 2 / (0.816 * 1.63) =.50 (note, there are several formulas that can be used to calculate r, see lecture slides for additional options) b = r * (s y / s x ) =.50 * (1.63 / 0.816) = 1.00 a = ȳ - bx = 3 (1)*(2) = 1 regression line is ŷ = 1 + 1*x

6 Calculate the correlation coefficient (r) and the best fitting regression line for the set of data below. To help, x = 2, s x = 1.41, ȳ = 3, and s y = 2.12: X Y (x x ) (y - ȳ) (x x )(y - ȳ) = = 3 (-1)(3) = = = -2 (2)(-2) = = = 1 (-1)(1) = = = 0 (-1)(0) = = = -2 (1)(-2) = -2 r = (1 / (n-1)) * (Ʃ(x - x )(y - ȳ))/ (s x * s y ) = (1 / (5-1)) * (-10) / (1.41 * 2.12) = (1/4) * (-10) / (1.41 * 2.12) = -.84 (note, there are several formulas that can be used to calculate r, see lecture slides for additional options) b = r * (s y / s x ) = -.84 * (2.12/ 1.41) = a = ȳ - bx = 3 (-1.26)*(2) = 5.52 regression line is ŷ = *x If you d like more practice doing this by hand, an option is to generate a small set of data, create vectors x and y in R to get the mean and standard deviations, do the rest of the calculations by hand, and then check your answers in R. Something like this: x <- c(1, 2, 3, 4, 5) y <- c(2, 3, 2, 4, 4) mean(x) sd(x) mean(y) sd(y) # stop, do calculations by hand # then check your work cor(x, y) # get correlation coefficient lm(y ~ x) # get regression coefficients (note that this call to lm() is working directly with vectors instead of with columns in a dataframe, and so the syntax does not include a dataframe) as an additional note, if you d like more practice calculating ŷ and residuals, you can check your work with model <- lm(y ~ x) fitted(model) # gives fitted y values, ŷ resid(model) # gives residuals Part 3: Correlation and regression with R (case studies) The remaining part of this problem set is in R (see P10_PracticePS_W10_R). This section of the problem set uses R to quickly generate correlation coefficients and regression equations from larger data sets, but primarily involves practice using this information to perform additional calculations and answer conceptual questions.