4B.7.2-4B.7.3 Paired and two-sample t-tests Quiz: Pearson Edexcel Further Maths, Unit 24

The 20 questions

1. For a paired t-test with n pairs and differences d, which statistic is used under H0?

   • t = d-bar / (s_d sqrt(n)), with 2n - 2 degrees of freedom
   • chi-squared = (n - 1) s_d^2 / sigma^2
   • z = d-bar / sigma, with known sigma
   • t = d-bar / (s_d / sqrt(n)), with n - 1 degrees of freedom

2. What is the pooled estimate of the common variance for two independent samples of sizes n1 and n2?

   • s_p^2 = (s1^2 + s2^2) / 2
   • s_p^2 = (n1 s1^2 + n2 s2^2) / (n1 + n2)
   • s_p^2 = ((n1 - 1) s1^2 + (n2 - 1) s2^2) / (n1 + n2)
   • s_p^2 = ((n1 - 1) s1^2 + (n2 - 1) s2^2) / (n1 + n2 - 2)

3. What are the degrees of freedom for the pooled two-sample t-test?

   • n1 + n2
   • min(n1, n2) - 1
   • n1 + n2 - 1
   • n1 + n2 - 2

4. Which conditions are required for the pooled two-sample t-test with equal but unknown variances?

   • Samples of equal size with different population variances
   • Samples from populations with unknown distributions and any variances
   • Independent samples from normal populations with equal population variances
   • Paired observations from populations with equal means

5. When is a paired t-test appropriate rather than a two-sample test?

   • When the observations are matched pairs, such as the same individuals measured twice
   • When the population variances are known
   • When the two samples have different sizes
   • When the two samples are large and independent with known variances

6. Why is a paired test usually more powerful than an unpaired test when the pairs are positively correlated?

   • Pairing avoids the need for the t distribution
   • Pairing doubles the sample size, so the test has twice as many observations
   • Pairing removes between-subject variation, which reduces the variance of the differences
   • Pairing increases the variance of the differences

7. What is a 95% confidence interval for mu1 - mu2 when the variances are equal but unknown?

   • (x-bar - y-bar) +/- t(n1 + n2 - 2) sqrt(s1^2/n1 + s2^2/n2)
   • (x-bar - y-bar) +/- t(n1 + n2 - 2) s_p sqrt(n1 + n2)
   • (x-bar - y-bar) +/- z s_p sqrt(1/n1 + 1/n2)
   • (x-bar - y-bar) +/- t(n1 + n2 - 2) s_p sqrt(1/n1 + 1/n2)

8. Five paired differences are 2, 4, 1, 3, 0. What is the t statistic for H0: mu_d = 0, to 2 decimal places?

   • 2.83
   • 0.71
   • 1.58
   • 2.00

9. Two independent samples have n1 = 8, s1 = 3 and n2 = 10, s2 = 4. What is the pooled variance s_p^2, to 2 decimal places?

   • 12.50
   • 12.94
   • 14.00
   • 11.50

10. Using s_p^2 = 12.94 with n1 = 8 and n2 = 10, what is the t statistic for a difference of means of 5, to 2 decimal places?

    • 5.00
    • 2.93
    • 1.71
    • 0.34

11. The statistic t = 2.93 has 16 df. The two-tailed 5% critical value is 2.120. What is the conclusion?

    • Accept H0, since the population variances are not equal
    • Reject H0 only if the statistic exceeds 5.00
    • Do not reject H0, since the samples differ in size
    • Reject H0, since 2.93 exceeds 2.120, so there is evidence the means differ

12. Ten paired differences have mean 2 and standard deviation 4. What is the 95% confidence interval for the mean difference, to 2 decimal places? (t value with 9 df is 2.262.)

    • (-2.00, 6.00)
    • (1.14, 2.86)
    • (0.00, 4.00)
    • (-0.86, 4.86)

13. Two independent samples have n1 = 10 with s1^2 = 4 and n2 = 20 with s2^2 = 9. What is the pooled variance s_p^2, to 2 decimal places?

    • 6.00
    • 8.25
    • 6.50
    • 7.39

14. A paired confidence interval for the difference (after minus before) is (-2.8, -0.2). What can be concluded?

    • The mean increased by 1.5, which is significant
    • Only the sign matters, and the test is not significant
    • There is no change, since the interval includes zero
    • The measurement tends to fall after the change, and since the interval excludes zero there is evidence of a change

15. Two independent samples each have n = 5, with mean difference 4, and s1 = s2 = 2. What is the 95% confidence interval for mu1 - mu2, to 2 decimal places? (t value with 8 df is 2.306.)

    • (1.08, 6.92)
    • (1.67, 6.33)
    • (2.00, 6.00)
    • (0.09, 7.91)

16. A researcher uses an unpaired two-sample test on data that are naturally paired and positively correlated. What is the consequence?

    • The standard error is smaller, so the test has higher power
    • The degrees of freedom double to 2(n - 1)
    • The standard error is larger than the paired analysis would give, so the test has lower power
    • The test becomes invalid because the data are not normal

17. Fifteen paired differences have mean 1.2 and standard deviation 2.0. What is the t statistic for H0: mu_d = 0, to 2 decimal places?

    • 0.62
    • 1.20
    • 2.32
    • 3.87

18. Two independent samples each have 10 observations, with sample standard deviations 2 and 8. Is the pooled t-test appropriate?

    • Appropriate, because equal sample sizes cancel the variance difference
    • Appropriate, since both samples are normal
    • Appropriate, but only if the two sample means are equal
    • Not appropriate, since the sample variances differ by a factor of 16, so the equal-variance assumption looks doubtful

19. Six paired differences are 3, 2, 4, 1, 5, 3. What is the t statistic for H0: mu_d = 0, to 2 decimal places?

    • 2.45
    • 3.00
    • 1.41
    • 5.20

20. Why does pooling the two sample variances give a more precise test?

    • Pooling doubles the degrees of freedom to 2(n - 1)
    • Combining the two variance estimates gives a better estimate of the common variance than either sample alone
    • Pooling removes the need for the populations to be normal
    • Pooling makes the two sample means equal

Related quizzes

• One-sample t-test · 4B.7.1 · 20 questions
• Proof by mathematical induction Quiz · 1.1 · 20 questions
• Quadratic equations and complex arithmetic Quiz · 2.1-2.2 · 20 questions
• Matrix arithmetic and inverses Quiz · 3.1-3.2 · 20 questions
• Expectation of discrete random variables Quiz · 3B.1.1 · 20 questions
• The Poisson distribution Quiz · 3B.2.1 · 20 questions
• Geometric and negative binomial models Quiz · 3B.3.1 · 20 questions
• Hypothesis tests for the Poisson distribution Quiz · 3B.4.1 · 20 questions
• Applying the Central Limit Theorem Quiz · 3B.5.1 · 20 questions
• Goodness of fit tests for discrete distributions Quiz · 3B.6.1 · 20 questions