Correlation & Regression – Pearson’s r, Covariance Quiz - ICSE Class 11 Mathematics

Free practice quiz: Correlation & Regression – Pearson’s r, Covariance (ICSE Class 11 Mathematics). 46 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This ICSE Class 11 Mathematics practice quiz has 46 multiple-choice questions on Correlation & Regression – Pearson’s r, Covariance, and 46 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

Sample questions

1. If the covariance between two random variables X and Y is 15 and their variances are $\sigma_X^2 = 25$ and $\sigma_Y^2 = 36$, what is the Pearson correlation coefficient (r)?

   1. 0.5
   2. 0.75
   3. 0.6
   4. 0.9
   Show answer

   Answer: A. 0.5

   The Pearson correlation coefficient (r) is calculated as $r = \frac{Cov(X, Y)}{\sigma_X \sigma_Y}$. Given $Cov(X, Y) = 15$, $\sigma_X = \sqrt{25} = 5$, and $\sigma_Y = \sqrt{36} = 6$. Therefore, $r = \frac{15}{5 \times 6} = \frac{15}{30} = 0.5$.

2. A dataset shows a Pearson correlation coefficient of -0.9. What does this imply?

   1. A weak positive linear relationship.
   2. A strong negative linear relationship.
   3. No linear relationship.
   4. A moderate negative linear relationship.
   Show answer

   Answer: B. A strong negative linear relationship.

   Pearson's r ranges from -1 to +1. A value close to -1 indicates a strong negative linear relationship, meaning as one variable increases, the other tends to decrease linearly. A value of -0.9 is close to -1, indicating a strong negative association.

3. Consider two variables X and Y. If $Cov(X, Y) = 0$, which of the following is NOT necessarily true?

   1. X and Y are linearly uncorrelated.
   2. The Pearson correlation coefficient is 0.
   3. X and Y are independent.
   4. The points on a scatter plot show no linear trend.
   Show answer

   Answer: C. X and Y are independent.

   A covariance of 0 means that X and Y are linearly uncorrelated and their Pearson correlation coefficient is 0. This suggests no linear trend in a scatter plot. However, it does not guarantee independence, as there might be a non-linear relationship between X and Y.

4. If $Cov(X, Y) = 30$ and $Cov(X, Z) = 20$, what is $Cov(X, Y+Z)$?

   1. 10
   2. 50
   3. 600
   4. 20
   Show answer

   Answer: B. 50

   Using the linearity property of covariance: $Cov(X, Y+Z) = Cov(X, Y) + Cov(X, Z)$. Substituting the given values: $Cov(X, Y+Z) = 30 + 20 = 50$.

5. If $Var(X) = 16$, $Var(Y) = 4$, and $Cov(X, Y) = 4$, calculate the Pearson correlation coefficient.

   1. 0.25
   2. 0.5
   3. 1
   4. 0.125
   Show answer

   Answer: B. 0.5

   First, find the standard deviations: $\sigma_X = \sqrt{Var(X)} = \sqrt{16} = 4$ and $\sigma_Y = \sqrt{Var(Y)} = \sqrt{4} = 2$. Then, use the formula for Pearson's r: $r = \frac{Cov(X, Y)}{\sigma_X \sigma_Y} = \frac{4}{4 \times 2} = \frac{4}{8} = 0.5$.

More questions in this quiz

Answers and explanations are in the full quiz.

1. Two variables X and Y have a perfect positive linear relationship. What is the value of Pearson's r?
2. If the Pearson correlation coefficient between two variables is 0, what can be concluded about their relationship?
3. The covariance between the scores of two students in two different subjects is -20. The standard deviation of the first student's scores is 5 and the second student's scores is 8. What is the Pearson correlation coefficient between their scores?
4. Which of the following statements about covariance is FALSE?
5. If $Cov(X, Y) = 0$, which of the following statements must be true?
6. If the variances of two variables X and Y are $\sigma_X^2 = 16$ and $\sigma_Y^2 = 9$, and their Pearson correlation coefficient is $r = 0.75$, what is their covariance?
7. Which statement best describes the relationship between covariance and Pearson's r?
8. If we have data points $(x_1, y_1), (x_2, y_2), ..., (x_n, y_n)$, and we transform the data by $x'_i = x_i - c$ and $y'_i = y_i - d$, where c and d are constants. How does the covariance $Cov(X', Y')$ relate to $Cov(X, Y)$?
9. If $Cov(X, Y) = 25$, $\sigma_X = 10$, and $\sigma_Y = 8$, what is the Pearson correlation coefficient?
10. Consider the transformation $X' = aX$ and $Y' = bY$, where a and b are positive constants. How does $Cov(X', Y')$ relate to $Cov(X, Y)$?
11. What is the maximum possible value for Pearson's r?
12. If two variables are independent, what can be said about their covariance?
13. Given the data points (1, 2), (2, 4), (3, 6), (4, 8). What is the Pearson correlation coefficient?
14. If $Cov(X, Y) = -15$, $\sigma_X = 6$, and $\sigma_Y = 5$, calculate the Pearson correlation coefficient (r).
15. What is the minimum possible value for Pearson's r?
16. If $Var(X) = 100$ and $Var(Y) = 64$, and $Cov(X, Y) = 40$, what is the Pearson correlation coefficient?
17. Suppose the sample covariance between two variables X and Y is 12, and their sample standard deviations are $s_X = 5$ and $s_Y = 6$. What is the sample Pearson correlation coefficient?
18. If $X$ and $Y$ are random variables, which of the following properties is NOT necessarily true for covariance $Cov(X, Y)$?
19. If the Pearson correlation coefficient between two variables is 0.8, what can be inferred about their relationship?
20. What is the effect of multiplying one of the variables by a negative constant on the covariance?
21. Two variables X and Y have a Pearson correlation coefficient of -0.7. This implies:
22. If $Cov(X, Y) = 0$ and $Var(X) > 0$ and $Var(Y) > 0$, what is the Pearson correlation coefficient?
23. Consider the data: X = {1, 2, 3}, Y = {3, 2, 1}. What is the covariance between X and Y?
24. If the Pearson correlation coefficient between two variables is close to 0, it implies:
25. Which of the following is a true statement about Pearson's r?
26. If $Cov(X, Y) = -24$, $\sigma_X = 6$, and $\sigma_Y = 5$, what is the Pearson correlation coefficient?
27. What is the relationship between $Cov(aX, Y)$ and $Cov(X, Y)$ where 'a' is a constant?
28. If a dataset exhibits a perfect negative linear relationship, what is the value of Pearson's r?
29. If $Cov(X, Y) = 15$ and $Cov(Y, X) = 15$, what can be inferred about the relationship between $Cov(X, Y)$ and $Cov(Y, X)$?
30. Consider a scatter plot where the points form a clear upward trend. What is the likely value of Pearson's r?
31. If $Cov(X, Y) = -10$ and $Cov(X, Z) = 5$, what is $Cov(X, Y-Z)$?
32. Which of the following statements about covariance is correct?
33. If X and Y are statistically independent, which of the following statements is necessarily true?
34. What does a Pearson correlation coefficient of $r=0$ imply?
35. If $Cov(X, Y) = 24$, $\sigma_X = 8$, and $\sigma_Y = 5$, calculate the Pearson correlation coefficient.
36. Consider the transformation $X' = X + c$ and $Y' = Y + d$, where c and d are constants. How does $Cov(X', Y')$ relate to $Cov(X, Y)$?
37. If the Pearson correlation coefficient between two variables is close to -1, what does this indicate?
38. If $Cov(X, Y) = 0$, $Var(X) = 10$, and $Var(Y) = 20$, what is the Pearson correlation coefficient?
39. What is the minimum possible value for the covariance between two variables?
40. If $Cov(X, Y) = 10$, $Var(X) = 25$, and $Var(Y) = 16$, calculate the Pearson correlation coefficient.
41. If $Cov(X, Y) = 30$, $Var(X) = 36$, $Var(Y) = 49$. Calculate Pearson's r.

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