Flashcards · 4040 · Topic 9

Bivariate distributions and their representation by scatter diagrams flashcards

11 cards on Topic 9, subtopics 9.1 to 9.2. Pick a subtopic or study them all.

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What do positive and negative correlation look like?: Positive: y tends to increase as x increases, so the points slope up. Negative: y tends to decrease as x increases, so the points slope down.
What decides the strength of a correlation?: How close the points lie to a straight line: strong if they hug a line, weak if widely scattered but still showing a trend. Strength is about scatter, not about how steep the line is.
How do you describe a correlation for full marks?: Give the type, the strength and the meaning in context, for example "weak negative correlation: people who watch more television tend to score slightly lower".
Does correlation show that x causes y?: No, it shows an association only. A lurking variable may drive both: ice cream sales and sunburn cases are both driven by hot weather.
What is an outlier on a scatter diagram?: A point far from the general pattern. It may pull a line of best fit, so check whether it is an error and comment on it.

How do you find a line of best fit by the method of semi-averages?: Order the pairs by x and split into two equal halves (for an odd number, leave out the middle pair and say so). Find the mean point of each half, then the gradient and intercept of the line through them.
Price x: 2, 4, 6, 8, 10, 12; sold y: 30, 27, 24, 19, 15, 11. Find the semi-average line: Mean points (4, 27) and (10, 15). m = (15 − 27) ÷ (10 − 4) = −2. 27 = −8 + c, so c = 35. y = −2x + 35.
How do you find the equation of a line through (2, 7) and (10, 19)?: m = (19 − 7) ÷ (10 − 2) = 1.5. c = 7 − 1.5 × 2 = 4. So y = 1.5x + 4.
With 7 pairs, x = 1, 3, 5, 7, 9, 11, 13 and y = 5, 9, 10, 17, 24, 27, 33, find the semi-average line: Leave out (7, 17). Mean points (3, 8) and (11, 28). m = 20 ÷ 8 = 2.5. 8 = 7.5 + c, so c = 0.5. y = 2.5x + 0.5.
Why is a prediction from y = −2x + 35 at x = 14 less reliable than at x = 7, if x ran from 2 to 12?: x = 7 is inside the data (interpolation, y = 21). x = 14 is outside it (extrapolation, y = 7), and the pattern may not continue. The line even gives y = 0 at x = 17.5, and sales cannot be negative.
Does the semi-average line pass through the mean point of the data?: Yes, when the two halves together contain all the data (an even number of pairs). The mean point is (mean of all x, mean of all y).