Bivariate Statistics

A scatter plot shows that as $x$ increases, $y$ tends to $decrease$. Describe the correlation.

A scatter plot shows random points with no clear trend. Describe the correlation.

Points lie tightly along a straight line going up. Describe the strength and direction of correlation.

A study plots height (cm) against weight (kg). Which is the (a) independent variable and (b) dependent variable, by convention if predicting weight?

A graph shows hours of sunshine vs ice cream sales. The points trend upward. Describe the relationship.

A line of best fit has equation $y = 2x + 10$. Predict $y$ when $x = 8$.

A line of best fit has equation $y = -2x + 15$. State the gradient and $y$-intercept.

A study finds a positive correlation between ice cream sales and shark attacks. Does eating ice cream cause shark attacks?

In a scatter of test scores vs revision hours, one point sits well above the line of best fit. What does this represent?

A regression line of $y$ on $x$ is $y = 0.8 x + 5$. Predict $y$ for $x = 20$.

A line of best fit passes through $(0, 4)$ and $(6, 22)$. Find its equation.

A regression line for height ($y$, cm) on age ($x$, years) is $y = 6x + 70$. Predict the height of a 9-year-old.

A regression line of test mark ($y$) on hours of revision ($x$) is $y = 5x + 40$. Interpret the gradient in context.

A regression line of test mark ($y$) on hours of revision ($x$) is $y = 5x + 40$. Interpret the $y$-intercept in context.

For each correlation coefficient $r$, describe the correlation. $r = -0.42$.

A line of best fit is $y = 3x + 5$. Find $x$ when $y = 32$.

A regression line of $y$ on $x$ passes through the mean point $(\bar{x}, \bar{y})$. If $\bar{x} = 8$ and the line is $y = 2x + 6$, find $\bar{y}$.

A regression equation was fitted using $x$-values from 5 to 20. Using the line, $y$ is predicted for $x = 10$ (case A) and $x = 30$ (case B). Which is interpolation, which is extrapolation?

Plot 1 has $r = 0.9$ and Plot 2 has $r = 0.2$. Which shows a stronger linear relationship?

A scatter plot of time spent gaming vs test scores has line of best fit $y = -2x + 70$. Interpret in context.

Five data points: (1, 3), (2, 5), (3, 7), (4, 9), (5, 11). Find the regression line by inspection.

A regression line $y = 0.5x + 20$ was fitted to $x$-values from 10 to 50. (a) Predict $y$ at $x = 30$. (b) At $x = 80$ the prediction would be?

A car's mileage (km) vs its resale value (£). Suggested data has $r \approx -0.92$. Interpret.

A regression line has a strong outlier. Will removing it (a) increase or decrease $|r|$? (b) Why?

Five data points: $\bar{x} = 5, \bar{y} = 12$; regression line has gradient $3$. State the regression equation.

Categorise the correlation based on $r$: (a) $r = 0.3$, (b) $r = 0.7$, (c) $r = -0.95$, (d) $r = 0.05$.

Five data points have $\bar{x} = 4$ and $\bar{y} = 11$. Four are $(2, 7), (3, 9), (5, 13), (6, 15)$. Find the fifth.

Studies find $r = 0.9$ between countries' chocolate consumption and number of Nobel Prize winners. Does eating chocolate cause Nobel Prizes?

LOBF $y = 2x + 1$ predicts $y$ at $x = 5$. Actual measured value is 9. What is the residual?

A regression line predicts temperature in °C from altitude (m): $T = -0.006a + 15$. Predict $T$ at altitude 2000 m.

A regression has $r = 0.8$. What proportion of variation in $y$ is explained by $x$?

Two studies of "$y$ vs $x$" have regression lines: Study A: $y = 3x + 5$, $r = 0.6$. Study B: $y = 3x + 5$, $r = 0.9$. Which is a better fit? Why?

A regression line $y = 2x + 5$ has $r^2 = 0.7$. At $x = 10$, you predict $y = 25$. What does $r^2 = 0.7$ tell you?

Three points and their LOBF predictions: $(1, 4) \to$ predicted 3.5; $(2, 6) \to$ predicted 5.5; $(3, 8) \to$ predicted 7.5. Find the sum of residuals.

Five students rank their preference for two subjects. The rank correlation is 0.8. Interpret.

A scatter plot looks like a U-shape. A student fits a straight LOBF and finds $r = 0$. What does this tell us about the relationship?

A regression line of "ice cream sales" on "temperature (°C)" gives $y = 2.5x + 5$ for $x$ in $[15, 35]$. At $x = -10$°C, the prediction is $-20$. Comment.

A dataset has $r = -0.7$ and $r^2 = 0.49$. Write a one-sentence summary for a non-technical audience.

For a sample of cars, fuel economy ($y$, mpg) vs engine size ($x$, L) gives $y = -8x + 50$. (a) Predict mpg for a 2 L engine. (b) Comment on $x = 5$ L.

A study finds $r = 0.95$ between heights of parent and child. Does this mean a tall parent guarantees a tall child?