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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.

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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.
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    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.
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    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
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    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: $r = \frac{Cov(X, Y)}{\sigma_X \sigma_Y} = \frac{4}{4 \times 2} = \frac{4}{8} = 0.5$.

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