Lesson 4B.7.1

4B.7.1 One-sample t-test: Pearson Edexcel Further Maths, Unit 24

20 questions

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Lesson 4B.7.1, One-sample t-test: 20 multiple choice questions for the Pearson Edexcel Further Maths (9FM0), Unit 24: t-tests, written with Revision Ninja.

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The 20 questions

  1. What is the test statistic for H0: mu = mu0 for a one-sample t-test with unknown sigma?

    • (x-bar - mu0) / s
    • (x-bar - mu0) / (s sqrt(n))
    • (x-bar - mu0) / (s / sqrt(n))
    • (x-bar - mu0) / (sigma / sqrt(n))

  2. Which assumption is needed for the one-sample t-test to be valid for small samples?

    • The population is discrete
    • The population variance is known
    • The sample must be paired
    • The population is approximately normal

  3. Why does the t distribution have heavier tails than the standard normal for small samples?

    • The t distribution is used only for paired data
    • The t distribution is always twice as wide as the standard normal
    • Estimating sigma by s adds extra uncertainty, so the distribution is more spread out when the degrees of freedom are few
    • The t distribution has a larger mean than the normal

  4. What is a 95% confidence interval for mu when sigma is unknown?

    • x-bar +/- t(n-1) sigma / sqrt(n)
    • x-bar +/- t(n) s / n
    • x-bar +/- s / sqrt(n-1)
    • x-bar +/- t(n-1) s / sqrt(n), with the 95% t critical value

  5. What is the two-tailed 5% critical t value with 15 degrees of freedom?

    • 1.753
    • 2.947
    • 2.602
    • 2.131

  6. For H1: mu > mu0, which rejection region is used?

    • Both tails equally, with half the significance level in each
    • The lower tail only, where t is less than the negative critical value
    • The upper tail, where t is greater than the positive critical value
    • The central region between minus and plus the critical value

  7. What is meant by the significance level of a hypothesis test?

    • The proportion of the sample lying above the mean
    • The probability that H0 is true
    • The probability of rejecting H0 when H0 is actually true
    • The sample size needed to achieve full power

  8. A sample of n = 16 has mean 52 and s = 6. Test H0: mu = 50 against H1: mu not equal to 50. What is the test statistic?

    • 1.50
    • 1.33
    • 0.33
    • 3.00

  9. For the statistic t = 1.33 with 15 df, what is the conclusion at the 5% two-tailed level? (Critical value 2.131.)

    • Reject H0, since t is positive so the test is one-tailed
    • Reject H0, since 1.33 exceeds 1.753
    • Do not reject H0, since |t| = 1.33 is below 2.131
    • The test cannot be carried out because the sample is too small for t

  10. A sample of n = 25 has mean 10.4 and s = 2. Test H0: mu = 10. What is the test statistic?

    • 2.0
    • 1.0
    • 0.2
    • 5.0

  11. Using the sample in the previous setting (n = 25, mean 10.4, s = 2), what is the 95% confidence interval for the mean, to 2 decimal places? (t value with 24 df is 2.064.)

    • (8.40, 12.40)
    • (9.57, 11.23)
    • (9.97, 10.83)
    • (9.62, 11.18)

  12. A sample of n = 9 has mean 23 and s = 4. Test H0: mu = 20 against H1: mu > 20 at the 5% level. What is the test statistic?

    • 0.75
    • 6.00
    • 2.25
    • 1.33

  13. A sample of n = 10 has mean 5 and s = 2. What is the half-width of the 95% confidence interval for the mean, to 2 decimal places? (t value with 9 df is 2.262.)

    • 1.43
    • 0.63
    • 1.96
    • 2.26

  14. A small sample comes from a strongly skewed population. How reliable is the one-sample t-test?

    • Less reliable, so normality should be checked or a non-parametric alternative considered
    • Valid only if the population variance is known
    • Perfectly valid regardless of the population shape
    • Always valid whenever n is at least 2

  15. A sample of n = 36 has mean 47 and s = 6. Test H0: mu = 50 against H1: mu not equal to 50. What is the test statistic?

    • -3.00
    • -6.00
    • 3.00
    • -0.50

  16. For a test statistic t = 2.5 with 20 df, what is the two-tailed p-value, to 2 decimal places?

    • 0.02
    • 0.01
    • 0.10
    • 0.05

  17. A sample of n = 17 has mean 50 and s = 8. What is the 95% confidence interval for the mean, to 2 decimal places? (t value with 16 df is 2.120.)

    • (46.00, 54.00)
    • (48.04, 51.96)
    • (45.89, 54.11)
    • (42.00, 58.00)

  18. A sample of n = 10 has mean 62 and s = 8. Test H0: mu = 60 against H1: mu not equal to 60 at the 5% level. What is the conclusion? (Critical value 2.262 with 9 df.)

    • Cannot be concluded, since the population standard deviation is unknown
    • Significant at 5%, since t is about 2.53
    • Significant, since 62 is greater than 60
    • Not significant, since t of about 0.79 is below 2.262

  19. Why do the degrees of freedom matter for the t critical value?

    • Fewer degrees of freedom make s a less precise estimate of sigma, so the critical value is larger
    • Degrees of freedom equal n, so no adjustment to the critical value is needed
    • Larger degrees of freedom give heavier tails, so the critical value is larger
    • Degrees of freedom count the parameters, so t is always symmetric about the mean

  20. A test gives a two-tailed p-value of 0.04 and a student concludes that the population mean differs from mu0 with 99% certainty. Which evaluation is correct?

    • Incorrect, because a two-tailed p-value is always smaller than a one-tailed p-value
    • Incorrect: the p-value is not the probability that H0 is true, and significance at 5% does not imply 99% certainty
    • Correct: a p-value of 0.04 means there is a 4% chance that mu equals mu0 exactly
    • Correct: a p-value below 0.05 implies 99% certainty of the conclusion

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