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 109 questions on this lesson, each with an explanation.
The strongest inverse correlation coefficient of the following is ……
Correct answer: (A) −0.84
Explanation: Step 1: An inverse correlation has a negative coefficient, and the strongest is the one nearest to −1. Step 2: Among −0.68, −0.35 and −0.84, the nearest to −1 is −0.84. ∴ −0.84 (0.92 is a trap: it is stronger in size but it is a direct correlation, not an inverse one).
If the number of pairs of values is n = 8 and the sum of the squares of the differences between the ranks is ΣD² = 21, then Spearman's rank correlation coefficient = ……
Correct answer: (A) 0.75
Explanation: Step 1: r = 1 − 6ΣD² ÷ (n(n² − 1)) = 1 − 6 × 21 ÷ (8 × (64 − 1)). Step 2: = 1 − 126 ÷ 504 = 1 − 0.25 = 0.75. ∴ r = 0.75 (0.25 is a trap: it forgets to subtract from 1; 0.9583 forgets the 6; 0.7539 uses n³ instead of n(n² − 1)).
If the correlation coefficient between two variables x and y equals −1, then the correlation between them is ……
Correct answer: (B) perfect inverse
Explanation: Step 1: r = −1 means perfect inverse correlation, r = 1 means perfect direct correlation and r = 0 means the correlation is nihilistic (no correlation). ∴ The correlation is perfect inverse.
If the ranks of five students in statistics are 1, 2, 3, 4, 5 and their corresponding ranks in mathematics are 2, 1, 4, 3, 5, then Spearman's rank correlation coefficient = ……
Correct answer: (C) 0.8
Explanation: Step 1: The differences D are −1, 1, −1, 1, 0, so ΣD² = 1 + 1 + 1 + 1 + 0 = 4. Step 2: r = 1 − 6 × 4 ÷ (5 × 24) = 1 − 24 ÷ 120 = 1 − 0.2 = 0.8. ∴ r = 0.8 (0.2 is a trap: it forgets to subtract from 1; 0.96 forgets the 6).
If n = 5, Σx = 30, Σy = 40, Σx² = 220, Σy² = 360 and Σxy = 266, then Pearson's correlation coefficient between x and y, to two decimal places, = ……
Correct answer: (B) 0.65
Explanation: Step 1: Numerator = nΣxy − Σx Σy = 5 × 266 − 30 × 40 = 1330 − 1200 = 130. Step 2: Denominator = √(5 × 220 − 30²) × √(5 × 360 − 40²) = √200 × √200 = 200. ∴ r = 130 ÷ 200 = 0.65, a direct correlation (0.46 is a trap: it comes from using Σx² and Σy² instead of nΣx² − (Σx)² and nΣy² − (Σy)² under the roots).
The price (x) of six books and their sales (y) were: price: Low, High, Average, Very high, Very low, High — sales, in the same order: High, Low, High, Very low, Very high, Average. Then Spearman's rank correlation coefficient, to four decimal places, = ……
Correct answer: (B) −0.9143
Explanation: Step 1: Rank in ascending order (Very low = 1); equal levels take the mean of their places. Ranks of x: 2, 4.5, 3, 6, 1, 4.5; ranks of y: 4.5, 2, 4.5, 1, 6, 3. Step 2: D² = 6.25, 6.25, 2.25, 25, 25, 2.25, so ΣD² = 67. ∴ r = 1 − 6 × 67 ÷ (6 × 35) = 1 − 402 ÷ 210 ≈ −0.9143, an inverse correlation (0.681 is a trap: it forgets the 6; −0.8857 gives equal levels consecutive ranks).
The grades of seven students in two subjects were: first subject: Good, Excellent, Pass, Good, Weak, Very good, Pass — second subject, in the same order: Very good, Very good, Good, Pass, Weak, Excellent, Weak. Then Spearman's rank correlation coefficient, to four decimal places, = ……
Correct answer: (D) 0.8214
Explanation: Step 1: Rank in ascending order (Weak = 1); equal grades take the mean of their places. Ranks of the first subject: 4.5, 7, 2.5, 4.5, 1, 6, 2.5. Step 2: Ranks of the second subject: 5.5, 5.5, 4, 3, 1.5, 7, 1.5. Step 3: D² = 1, 2.25, 2.25, 2.25, 0.25, 1, 1, so ΣD² = 10. ∴ r = 1 − 6 × 10 ÷ (7 × 48) = 1 − 60 ÷ 336 ≈ 0.8214, a direct correlation (0.7857 is a trap: it gives equal grades different consecutive ranks).
The points (1, 9), (2, 7), (3, 6), (4, 3) represent the values of the two variables x and y. Which of the following changes to only one value of y makes Pearson's correlation coefficient between x and y equal to −1?
Correct answer: (A) Changing 6 to 5
Explanation: Step 1: r = −1 means that all the points lie on a straight line with a negative slope. Step 2: The points (1, 9), (2, 7), (4, 3) lie on the line y = 11 − 2x, and at x = 3 this line gives y = 5. ∴ Changing 6 to 5 makes the correlation perfect inverse; every other change leaves a point off the line (for example (1, 8), (2, 7), (3, 6) lie on a line of slope −1 but (4, 3) does not), so r is between −1 and 0.
The values of x are 1, 2, 3, 4, 5, 6 and the corresponding values of y are 2, 3, 5, 6, 8, 1. If the pair (6, 1) is deleted from the data, then Pearson's correlation coefficient between x and y, to four decimal places, changes from …… to ……
Correct answer: (D) 0.2228 to 0.9934
Explanation: Step 1: Before: n = 6, Σx = 21, Σy = 25, Σx² = 91, Σy² = 139, Σxy = 93, so r = (558 − 525) ÷ (√105 × √209) ≈ 0.2228 (a weak direct correlation). Step 2: After: n = 5, Σx = 15, Σy = 24, Σx² = 55, Σy² = 138, Σxy = 87, so r = (435 − 360) ÷ (√50 × √114) ≈ 0.9934 (a strong direct correlation). ∴ r changes from 0.2228 to 0.9934: the outlying point (6, 1) was weakening the correlation a lot.
In a study of two groups: first group: n = 4, Σx = 10, Σy = 10, Σx² = 30, Σy² = 30, Σxy = 28 — second group: n = 4, Σx = 26, Σy = 34, Σx² = 174, Σy² = 294, Σxy = 221. If the two groups are combined into one group, then Pearson's correlation coefficient of the combined data, to three decimal places, = ……
Correct answer: (B) 0.869
Explanation: Step 1: When the groups are combined the sums are added: n = 8, Σx = 36, Σy = 44, Σx² = 204, Σy² = 324, Σxy = 249. Step 2: r = (8 × 249 − 36 × 44) ÷ (√(1632 − 1296) × √(2592 − 1936)) = 408 ÷ (√336 × √656). ∴ r ≈ 0.869 (the first group's coefficient is 0.6 and the second's is 0; 0.3 is a trap: it is their mean, but the coefficient of combined data cannot be found from the two coefficients).
The first 10 questions in every subject are free, each with an explanation.