Lesson 4B.5.5

4B.5.5 Large sample confidence intervals Quiz: Pearson Edexcel Further Maths, Unit 22

20 questions

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Lesson 4B.5.5, Large sample confidence intervals: 20 multiple choice questions for the Pearson Edexcel Further Maths (9FM0), Unit 22: Confidence intervals, written with Revision Ninja.

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The 20 questions

  1. In a large-sample confidence interval for a mean with unknown sigma, what is the usual change?

    • Replace n with n - 1 in the standard error
    • Replace the interval with the range divided by 4
    • Replace sigma with the sample standard deviation s
    • Replace the mean with the median

  2. What does the central limit theorem state about the sample mean for large n?

    • The sample mean is exactly uniform for any population
    • The sample mean is only defined for discrete populations
    • The sample mean is approximately normal, whatever the shape of the population with finite variance
    • The sample mean is normal only if the population is normal

  3. What is the large-sample 95% confidence interval for a population mean with unknown sigma?

    • x-bar +/- s / sqrt(n), with no multiplier
    • x-bar +/- 1.96 s / sqrt(n)
    • x-bar +/- t s / n
    • x-bar +/- 1.96 s sqrt(n)

  4. What is the large-sample 95% confidence interval for mu_x - mu_y with unknown variances?

    • (x-bar - y-bar) +/- 1.96 (s_x^2 - s_y^2)/(n_x + n_y)
    • (x-bar - y-bar) +/- 1.96 sqrt(s_x^2/n_x + s_y^2/n_y)
    • (x-bar - y-bar) +/- 1.96 (s_x + s_y)/(n_x + n_y)
    • (x-bar - y-bar) +/- 1.96 sqrt((s_x^2 + s_y^2)/(n_x + n_y))

  5. What rule of thumb is commonly used to judge whether a sample is large enough for the z approximation?

    • A sample size of at least 100 in every case
    • Any sample size will do
    • A sample size below 10
    • A sample size of about 30 or more

  6. When sigma is unknown and n is large, what distribution is approximately followed by the standardised statistic?

    • The chi-squared distribution
    • The standard normal, N(0, 1)
    • The t distribution with n - 1 degrees of freedom
    • The uniform distribution

  7. For a large sample with unknown sigma, is the t distribution needed?

    • No, since z is only valid for paired data
    • Yes, since t is always used for large samples
    • Not necessarily, since for large n the t distribution is close to normal and z is an acceptable approximation
    • Yes, since z is never valid when sigma is unknown

  8. A sample of n = 100 has mean 40 and s = 8. What is the 95% confidence interval for the population mean, to 2 decimal places?

    • (38.43, 41.57)
    • (36.08, 43.92)
    • (38.00, 42.00)
    • (39.22, 40.78)

  9. A sample of n = 64 has mean 75 and s = 12. What is the 90% confidence interval for the mean, to 2 decimal places?

    • (70.00, 80.00)
    • (72.06, 77.94)
    • (73.50, 76.50)
    • (72.53, 77.47)

  10. Sample A: n = 60, mean 9 with s = 5. Sample B: n = 80, mean 5 with s = 7. What is the 95% confidence interval for the difference of means, to 2 decimal places?

    • (1.00, 7.00)
    • (0.02, 7.98)
    • (3.00, 5.00)
    • (2.01, 5.99)

  11. A sample of n = 100 has s = 8. What is the standard error of the sample mean?

    • 0.8
    • 8
    • 0.08
    • 1.6

  12. A sample of n = 100 has s = 10. What is the width of the 95% confidence interval for the mean?

    • 19.6
    • 1.96
    • 0.98
    • 3.92

  13. A sample of n = 49 has mean 20 and s = 3.5. What is the 99% confidence interval for the mean, to 2 decimal places?

    • (19.02, 20.98)
    • (16.50, 23.50)
    • (18.71, 21.29)
    • (18.00, 22.00)

  14. A t distribution with 399 degrees of freedom has a two-tailed 95% critical value of about 1.97. Why is the z value of 1.96 an acceptable approximation here?

    • The t distribution with many degrees of freedom is almost identical to the standard normal
    • The t distribution has a larger variance than the normal for any sample size
    • The z value is always exact, and the t value is only an approximation
    • The degrees of freedom have no effect on the critical value

  15. Two independent samples each have sample standard deviation 3 and 4, with n = 50 in both groups. What is the standard error of the difference of means, to 2 decimal places?

    • 1.00
    • 0.50
    • 0.71
    • 0.25

  16. Which statement about using s in place of sigma for a large sample is correct?

    • It is exact because s equals sigma for large samples
    • It requires the population to be uniform
    • It is exact for any sample size when the data are normal
    • It is an approximation whose accuracy improves as n grows

  17. What is the smallest whole sample size n for a 95% confidence interval with half-width at most 0.5, when the sample standard deviation is about 6?

    • 24
    • 553
    • 144
    • 554

  18. A sample of n = 36 from a skewed population has mean 50 and s = 9. Is the large-sample z interval reliable?

    • No, because s must equal sigma exactly
    • Yes, exactly, because n = 36 makes t equal to z
    • Approximately yes, since the central limit theorem applies, though skewness calls for some caution at smaller n
    • No, because the central limit theorem requires a normal population

  19. Two independent samples each have n = 100 and s = 10 for both groups. What is the half-width of a 95% confidence interval for the difference of means, to 2 decimal places?

    • 1.41
    • 1.96
    • 2.77
    • 3.92

  20. For large samples the difference of means is -2.5 with a standard error of 1. What is the two-tailed p-value, to 3 decimal places?

    • 0.006
    • 0.012
    • 0.994
    • 0.025

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