Lesson 4B.6.1

4B.6.1 Hypothesis test for a population variance Quiz: Pearson Edexcel Further Maths, Unit 23

20 questions

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Lesson 4B.6.1, Hypothesis test for a population variance: 20 multiple choice questions for the Pearson Edexcel Further Maths (9FM0), Unit 23: Chi-squared variance test, written with Revision Ninja.

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The 20 questions

  1. Under H0: sigma^2 = sigma0^2, what is the test statistic for a sample of size n with sample variance s^2?

    • n s^2 / sigma0^2
    • (n - 1) s^2 / sigma0^2
    • (n - 1) sigma0^2 / s^2
    • (n - 1) s / sigma0

  2. In a chi-squared test of a population variance, what is a Type I error?

    • Using a t distribution instead of a chi-squared distribution
    • Rejecting H0 when the sample variance is zero
    • Rejecting H0 when H0 is actually true
    • Accepting H0 when H0 is actually false

  3. Which assumption is required for the chi-squared test of a single population variance?

    • The population is discrete
    • The sample size exceeds 100
    • The population is uniform
    • The population is normal

  4. What is the null hypothesis for the chi-squared variance test with known target sigma0?

    • H0: mu = mu0
    • H0: sigma^2 not equal to sigma0^2
    • H0: s^2 = sigma0^2 for the sample
    • H0: sigma^2 = sigma0^2

  5. For the alternative H1: sigma^2 > sigma0^2, when is H0 rejected?

    • When the test statistic falls below the lower critical value
    • When the test statistic exceeds the upper critical value
    • Whenever s^2 is smaller than sigma0^2
    • Only when the test statistic equals n - 1 exactly

  6. Which description of the chi-squared distribution is correct?

    • It is symmetric about zero
    • It is positively skewed and takes only non-negative values
    • It takes values only between 0 and 1
    • It is symmetric about its mean of n

  7. What does a 5% upper critical value for a chi-squared distribution mean?

    • The value equal to the population variance sigma0^2
    • The value equal to the sample variance s^2
    • The value that leaves 5% of the probability in the upper tail
    • The value that leaves 5% of the probability in the lower tail

  8. A sample of n = 10 has s^2 = 8. Test H0: sigma^2 = 4. What is the test statistic?

    • 20
    • 18
    • 5
    • 4.5

  9. The statistic in the previous setting is 18 with 9 degrees of freedom. The upper 5% critical value is 16.919. What is the conclusion for H1: sigma^2 > 4?

    • Reject H0, since s^2 = 8 is larger than the critical value
    • Accept H0, since 18 lies in the central 95% region of the distribution
    • Accept H0, since 18 is less than 19.023
    • Reject H0, since 18 exceeds 16.919, giving evidence that the variance exceeds 4

  10. A sample of n = 16 has s = 3. Test H0: sigma = 2 against H1: sigma > 2. What is the test statistic?

    • 6.67
    • 22.50
    • 33.75
    • 36.00

  11. For a test of H0: sigma = 2 against H1: sigma > 2 with statistic 33.75 and df 15, the upper 5% critical value is 25.00. What is the conclusion?

    • Do not reject H0, since 33.75 is less than 36.00
    • Do not reject H0, since n = 16 is too small for the chi-squared test
    • Reject H0, since 33.75 exceeds 25.00, so the standard deviation appears to exceed 2
    • Reject H0, since 33.75 is below the critical value 6.27

  12. A sample of n = 12 has s^2 = 5. Test H0: sigma^2 = 2 against H1: sigma^2 not equal to 2. What is the test statistic?

    • 2.27
    • 30.00
    • 13.75
    • 27.5

  13. When H0 is true, what is the expected value of the statistic (n - 1)s^2/sigma^2?

    • sigma^2
    • n - 1
    • 1
    • n

  14. A test of H1: sigma^2 < 4 uses a statistic of 3.2 with 9 degrees of freedom. The lower 5% critical value is 3.325. What is the conclusion?

    • Do not reject H0, since 3.2 is below 3.325
    • Do not reject H0, since 3.2 is above 2.700
    • Reject H0, since 3.2 exceeds 2.700
    • Reject H0, since 3.2 is below 3.325

  15. A sample of n = 20 has s^2 = 10. Test H0: sigma^2 = 4 against H1: sigma^2 > 4. The 1% upper critical value for 19 df is 36.19. What is the test statistic?

    • 19.0
    • 7.6
    • 47.5
    • 50.0

  16. A student says that a sample variance twice the hypothesised value always gives a significant result. Which evaluation is correct?

    • Yes, any s^2 larger than sigma0^2 is significant at the 5% level
    • Not always: the statistic depends on n - 1, so for a small sample a large s^2 may not be significant
    • No, because s^2 can never exceed sigma0^2 in a test
    • Yes, because the test statistic always equals s^2/sigma0^2

  17. A sample of n = 8 has s^2 = 6. Test H0: sigma^2 = 3 against H1: sigma^2 not equal to 3 at the 5% two-tailed level. Critical values for 7 df are 1.690 and 16.013. What is the conclusion?

    • Do not reject H0, since 14 is less than 2 times 7
    • Reject H0 at 5% one-tailed, since 14 exceeds 14.067
    • Do not reject H0, since 14 lies between 1.690 and 16.013
    • Reject H0, since 14 exceeds the lower critical value 1.690

  18. A sample of n = 10 has s^2 = 4. What is the 95% confidence interval for sigma^2, to 2 decimal places? (Chi-squared critical values for 9 df are 2.700 and 19.023.)

    • (4.00, 16.00)
    • (0.44, 3.33)
    • (1.89, 13.33)
    • (2.70, 19.02)

  19. For H1: sigma^2 > sigma0^2 with s^2/sigma0^2 fixed at 2, the statistic is (n - 1) x 2. By what factor does the statistic change when n increases from 10 to 40?

    • About 4.3, since 78/18 = 4.33
    • 2
    • 4
    • 16

  20. Why does the chi-squared variance test need two separate critical values for a two-tailed alternative?

    • The distribution is symmetric about n - 1, so one value serves both tails
    • Two values are needed only when n is even
    • Two values are needed because the degrees of freedom equal 2(n - 1)
    • The chi-squared distribution is not symmetric, so the lower and upper tails must be found separately

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