4B.3.3 Testing the product moment correlation coefficient Quiz: Pearson Edexcel Further Maths, Unit 20

The 20 questions

1. Which null hypothesis is used to test whether a product moment correlation coefficient is zero?

   • H0: sample r = 1
   • H0: rho = 0
   • H0: mu_x = mu_y
   • H0: rho = 1

2. Under what condition are the critical values for the product moment correlation coefficient valid?

   • Both variables must be discrete
   • The sample size must be at least 300
   • The data must first be converted to ranks
   • The data come from a population with a bivariate normal distribution

3. For the test statistic t = r sqrt(n - 2) / sqrt(1 - r^2), what are the degrees of freedom?

   • n - 2
   • n
   • n + 2
   • n - 1

4. Which test is suitable for testing correlation when the data are not believed to be bivariate normal?

   • The product moment correlation test using t critical values
   • Spearman's rank correlation test
   • The chi-squared test of a population variance
   • The paired t-test on the raw values

5. What happens to the product moment correlation r if a constant is added to every value of x?

   • r is multiplied by that constant
   • r is unchanged
   • r increases by that constant
   • r changes sign

6. A researcher expects a positive correlation and tests for it. Which alternative hypothesis is appropriate?

   • H1: rho < 0
   • H1: rho not equal to 0
   • H1: rho = 0
   • H1: rho > 0

7. In a hypothesis test for correlation, the p-value is less than the significance level. What is the correct conclusion?

   • Reject H0, and conclude there is evidence of a non-zero correlation
   • Accept H0 and conclude that rho is exactly zero
   • Prove that the population correlation equals the sample r
   • Conclude the data are bivariate normal

8. A sample of n = 12 pairs has r = 0.6. What is the test statistic t, to 2 decimal places?

   • 3.46
   • 1.58
   • 0.60
   • 2.37

9. Using the test statistic from a sample of n = 12 with r = 0.6, what is the conclusion at the 5% two-tailed level? (Critical value with 10 df is 2.228.)

   • The test cannot be carried out because n is too small
   • Reject H0, since r is greater than 0.5
   • Reject H0, since 2.37 exceeds 2.228
   • Accept H0, since 2.37 is less than 2.228

10. A sample of n = 20 pairs has r = 0.3. Is this significant at the 5% one-tailed level? (Critical value with 18 df is 1.734.)

    • Yes, since t = 1.33 exceeds 1.734
    • No, because samples must have at least 30 pairs
    • No, t = 1.33 is below the critical value 1.734
    • Yes, since r = 0.3 is positive

11. A sample of n = 10 pairs has r = -0.7. What is the test statistic t, to 2 decimal places?

    • 2.77
    • -0.91
    • -1.98
    • -2.77

12. A sample of n = 17 pairs has r = 0.45. What is the test statistic t, to 2 decimal places?

    • 1.74
    • 0.45
    • 1.95
    • 2.13

13. A sample of n = 8 pairs has r = 0.8. What is the test statistic t, to 2 decimal places?

    • 2.45
    • 4.80
    • 1.33
    • 3.27

14. A sample of n = 6 pairs has r = 0.9. Is H0 rejected at the 1% two-tailed level? (Critical value with 4 df is 4.604.)

    • Yes, since t = 4.13 exceeds 2.776
    • No, since t = 4.13 is below 4.604
    • No, because n must be at least 10
    • Yes, since r = 0.9 is above 0.5

15. A sample of n = 27 pairs has r = 0.4. What is the test statistic t, to 2 decimal places?

    • 1.09
    • 2.06
    • 0.40
    • 2.18

16. Why must the bivariate normal assumption hold for the product moment correlation test?

    • Normality guarantees that the mean equals the median for both variables
    • The critical values are derived from the distribution of r under bivariate normality, which the t-based test relies on
    • It guarantees that r equals 1 for large samples
    • It ensures the sample is random within each subgroup

17. A researcher has r = 0.45 from n = 12 pairs and tests H1: rho > 0 at the 5% level. (One-tailed critical value with 10 df is 1.812.) What is the conclusion?

    • Significant, because one-tailed tests are always more powerful
    • Not significant, since t = 1.59 is below 1.812
    • Not significant, because n must be at least 30
    • Significant, since r = 0.45 is above 0.4

18. A sample of pairs gives a large r but includes one extreme outlier. Why can the product moment test mislead?

    • Outliers affect only the means, not the value of r
    • One outlier can inflate or deflate r, and it can break the bivariate normal assumption behind the test
    • Outliers increase the degrees of freedom of the test
    • Outliers always increase r, so the test is always significant

19. A test gives t = 3.0 with 8 degrees of freedom. Two-tailed critical values are 2.306 (5%) and 3.355 (1%). What is the conclusion?

    • Not significant at either level
    • Significant at the 5% level but not at the 1% level
    • Significant at the 1% level but not at the 5% level
    • Significant at both the 5% and 1% levels

20. For the same sample correlation r = 0.3, which sample size gives the larger test statistic t?

    • The statistic depends only on the sign of r
    • Both give the same t
    • n = 100
    • n = 20

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