Unit 1 — Correlation and Regression · Statistics. These are 10 questions from the Magmo3 bank on this lesson, from easy to hard. Solve each one, then open "Show answer and explanation". Magmo3 has 117 questions on this lesson, each with an explanation.
If the regression line equation is ŷ = 12 + 0.75x, then when x increases by 8, the expected value of y increases by ……
Correct answer: (A) 6
Explanation: Step 1: An increase of 8 in x increases ŷ by b × 8 = 0.75 × 8 = 6; the constant 12 does not change. ∴ ŷ increases by 6 (18 is a trap: it wrongly adds the constant 12).
In the regression line of y on x, ŷ = a + bx, b is called ……
Correct answer: (B) the regression coefficient of y on x
Explanation: Step 1: b = (nΣxy − Σx Σy) ÷ (nΣx² − (Σx)²) is called the regression coefficient of y on x; it is the slope of the regression line. Step 2: a is the y-intercept, and ŷ is the expected (estimated) value of y. ∴ b is the regression coefficient of y on x.
If the regression coefficient of y on x equals −0.6, then the correlation between x and y is ……
Correct answer: (D) inverse
Explanation: Step 1: The regression coefficient b and the correlation coefficient r have the same numerator nΣxy − Σx Σy, and both denominators are positive. Step 2: So b and r have the same sign; b is negative. ∴ The correlation is inverse.
A company's sales (in thousand units) in the years 2019, 2020, 2021, 2022, 2023 were 12, 15, 17, 20, 21 respectively. Coding the years as x = 1, 2, 3, 4, 5 and finding the regression line of the sales y on x, the expected sales for 2024 = …… thousand units.
Correct answer: (B) 23.9
Explanation: Step 1: Σx = 15, Σy = 85, Σx² = 55, Σxy = 278. Step 2: b = (5 × 278 − 15 × 85) ÷ (5 × 55 − 15²) = (1390 − 1275) ÷ 50 = 2.3, and a = (85 − 2.3 × 15) ÷ 5 = 10.1. Step 3: The year 2024 has code x = 6: ŷ = 10.1 + 2.3 × 6 = 23.9. ∴ 23.9 thousand units (21.6 is a trap: it uses x = 5, the expected value for 2023; x = 7 belongs to 2025).
If the regression line equation of y on x is ŷ = 2 + 3x, Σx = 20 and Σy = 76, then the number of pairs of values n = ……
Correct answer: (C) 8
Explanation: Step 1: a = (Σy − bΣx) ÷ n, so 2 = (76 − 3 × 20) ÷ n = 16 ÷ n. ∴ n = 8 (68 is a trap: adding bΣx instead of subtracting it gives n = (76 + 60) ÷ 2).
In a first study: nΣxy − Σx Σy = 150, nΣx² − (Σx)² = 100, nΣy² − (Σy)² = 900, and in a second study: nΣxy − Σx Σy = 90, nΣx² − (Σx)² = 100, nΣy² − (Σy)² = 100. Then ……
Correct answer: (C) the regression coefficient of y on x is greater in the first, and the correlation is stronger in the second
Explanation: Step 1: First: b = 150 ÷ 100 = 1.5 and r = 150 ÷ (√100 × √900) = 150 ÷ 300 = 0.5. Step 2: Second: b = 90 ÷ 100 = 0.9 and r = 90 ÷ (√100 × √100) = 0.9. ∴ The slope is greater in the first, but the correlation is stronger in the second: a large regression coefficient does not mean a strong correlation.
If n = 5, Σx = 15, Σy = 20, Σx² = 55, Σy² = 86 and Σxy = 66, then Pearson's correlation coefficient r and the regression coefficient of y on x, b, are ……
Correct answer: (C) r ≈ 0.775, b = 0.6
Explanation: Step 1: The common numerator = nΣxy − Σx Σy = 5 × 66 − 15 × 20 = 30. Step 2: b = 30 ÷ (275 − 225) = 0.6. Step 3: r = 30 ÷ (√50 × √(430 − 400)) = 30 ÷ √1500 ≈ 0.775. ∴ r ≈ 0.775, b = 0.6 (2.2 is a trap: it is a, not b; both r and b are positive because the numerator is positive).
In two studies the values of x are 1, 2, 3, 4, 5. In the first, the corresponding values of y are 3, 5, 4, 8, 10, and in the second they are 10, 8, 4, 5, 3. Then the regression lines of y on x of the two studies intersect at the point ……
Correct answer: (A) (3, 6)
Explanation: Step 1: First: Σx = 15, Σy = 30, Σx² = 55, Σxy = 107, so b = (535 − 450) ÷ 50 = 1.7 and a = (30 − 25.5) ÷ 5 = 0.9. Step 2: Second: Σxy = 73, so b = (365 − 450) ÷ 50 = −1.7 and a = (30 + 25.5) ÷ 5 = 11.1. Step 3: 0.9 + 1.7x = 11.1 − 1.7x gives x = 3 and y = 6. ∴ (3, 6): it is the point (x̄, ȳ) shared by the two studies (x̄ = 3, ȳ = 6), and every regression line passes through the point of the means.
For five pairs of values: Σx = 20, Σy = 35, nΣx² − (Σx)² = 200, nΣy² − (Σy)² = 800, and Pearson's correlation coefficient between x and y = 0.75. Then the expected value of y when x = 6, using the regression line of y on x, = ……
Correct answer: (D) 10
Explanation: Step 1: From r: nΣxy − Σx Σy = 0.75 × √(200 × 800) = 0.75 × 400 = 300. Step 2: b = 300 ÷ 200 = 1.5 and a = (35 − 1.5 × 20) ÷ 5 = 1, so ŷ = 1 + 1.5x. Step 3: At x = 6: ŷ = 1 + 9 = 10. ∴ ŷ = 10 (8.5 is a trap: it takes b = r = 0.75; 7.75 divides by 800 instead of 200; 14 forgets to divide by n when finding a).
If n = 5, Σx = 15, Σy = 20, Σx² = 55, Σxy = 70, and a new variable z is formed where z = 2y + 3 for each value of y, then the regression line equation of z on x is ……
Correct answer: (D) ẑ = 5 + 2x
Explanation: Step 1: Σz = 2 × 20 + 3 × 5 = 55 and Σxz = 2 × 70 + 3 × 15 = 185. Step 2: b = (5 × 185 − 15 × 55) ÷ (275 − 225) = 100 ÷ 50 = 2 and a = (55 − 2 × 15) ÷ 5 = 5. ∴ ẑ = 5 + 2x (the line of y on x was ŷ = 1 + x; a = 2 is a trap: it forgets to add 3 × n to Σz, and keeping b = 1 is wrong because doubling y doubles the slope).
The first 10 questions in every subject are free, each with an explanation.