Lesson 4B.5.3-4B.5.4

4B.5.3-4B.5.4 Confidence limits for a mean and for a difference of means Quiz: Pearson Edexcel Further Maths, Unit 22

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson 4B.5.3-4B.5.4, Confidence limits for a mean and for a difference of means: 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. What is the confidence interval for a normal mean with sigma known, based on a sample of size n?

    • x-bar +/- z sigma / n
    • x-bar +/- sigma / sqrt(n), with no z value
    • x-bar +/- z sigma / sqrt(n)
    • x-bar +/- z sigma^2 / sqrt(n)

  2. What is the two-sided z value for a 95% confidence interval?

    • 1.282
    • 2.576
    • 1.96
    • 1.645

  3. What is the 95% confidence interval for the difference of two means with known variances sigma_x^2 and sigma_y^2?

    • (x-bar - y-bar) +/- z sqrt(sigma_x^2/n_x - sigma_y^2/n_y)
    • (x-bar - y-bar) +/- z sqrt((sigma_x^2 + sigma_y^2)/(n_x + n_y))
    • (x-bar - y-bar) +/- z sqrt(sigma_x^2/n_x + sigma_y^2/n_y)
    • (x-bar - y-bar) +/- z (sigma_x + sigma_y)/sqrt(n_x + n_y)

  4. For the test of H0: mu_x - mu_y = 0 with known variances, what is the distribution of the standardised test statistic under H0?

    • Chi-squared with n_x + n_y - 2 degrees of freedom
    • N(0, 2)
    • Standard normal, N(0, 1)
    • t with n_x + n_y - 2 degrees of freedom

  5. Which assumption is needed for the known-variance tests and intervals for means in this topic?

    • The variances must be unknown and estimated by a t-distribution
    • Only the means need to be known in advance
    • The samples must be paired observations
    • Both populations are normal with known variances, or the samples are large enough for the central limit theorem

  6. A population variance is known to be 16 and a sample of n = 64 is taken. What is the standard error of the sample mean?

    • 2
    • 0.25
    • 4
    • 0.5

  7. How does increasing the confidence level from 95% to 99% change a confidence interval for a mean, with n and sigma fixed?

    • It depends only on the sample size and not on the confidence level
    • It makes the interval wider
    • It leaves the width unchanged
    • It makes the interval narrower

  8. A sample of n = 25 from a population with sigma = 10 has mean 40. What is the 95% confidence interval for the population mean, to 2 decimal places?

    • (36.00, 44.00)
    • (38.00, 42.00)
    • (30.00, 50.00)
    • (36.08, 43.92)

  9. Using the same sample (n = 25, sigma = 10, mean 40), what is the 90% confidence interval for the mean, to 2 decimal places?

    • (36.71, 43.29)
    • (34.71, 45.29)
    • (37.00, 43.00)
    • (36.08, 43.92)

  10. Sample A: n = 40, mean 72, sigma = 8. Sample B: n = 50, mean 65, sigma = 6. What is the 95% confidence interval for mu_A - mu_B, to 2 decimal places?

    • (0.00, 14.00)
    • (4.01, 9.99)
    • (5.00, 9.00)
    • (6.00, 8.00)

  11. Using the samples in the previous setting (difference 7, standard error 1.523), what is the z statistic for H0: mu_A = mu_B, and the conclusion at the 5% two-tailed level?

    • z is about 4.60, so H0 is accepted because 4.60 is below 5
    • z is about 0.22, so H0 is accepted
    • z is about 4.60, so H0 is rejected: strong evidence that the population means differ
    • z is about 2.30, so H0 is rejected at the 1% level

  12. A sample of n = 16 from a population with sigma = 4 has mean 20. What is the 99% confidence interval for the mean, to 2 decimal places?

    • (17.42, 22.58)
    • (16.00, 24.00)
    • (18.04, 21.96)
    • (17.00, 23.00)

  13. To halve the width of a 95% confidence interval for a mean with sigma known, by what factor must the sample size n be multiplied?

    • 1/2
    • 2
    • 8
    • 4

  14. A 95% confidence interval for mu_x - mu_y is (1.2, 6.8). Does a two-tailed 5% test reject H0: mu_x = mu_y?

    • No, because 95% intervals never exclude zero
    • No, since the interval is centred at 4
    • Yes, since 6.8 is above 1.96
    • Yes, since 0 is not in the interval

  15. Samples have sigma_x = 3 with n_x = 9, and sigma_y = 4 with n_y = 16. What is the standard error of x-bar - y-bar, to 2 decimal places?

    • 2.00
    • 0.71
    • 1.41
    • 1.00

  16. What is the smallest whole sample size n for a 95% confidence interval for a mean with half-width at most 1, when sigma = 10?

    • 96
    • 97
    • 25
    • 49

  17. A 95% confidence interval for mu_x - mu_y is (-2.1, 0.5). Does a two-tailed 5% test reject H0: mu_x = mu_y?

    • It cannot be decided without the sample sizes
    • No, since 0 lies inside the interval
    • Yes, since the interval width exceeds 1
    • Yes, since most of the interval is negative

  18. Why is it valid to use the z value when sigma is known, for a sample mean from normal populations?

    • The z value applies only when the sample size is below 30
    • Sample variances are always equal to population variances
    • The z value is needed only when the samples are paired
    • The standardised mean then has exactly a standard normal distribution, so z applies

  19. Two samples each have n = 25 and sigma = 5. The difference of sample means is 3. What is the two-tailed p-value of the z test for equal means, to 3 decimal places?

    • 0.017
    • 0.034
    • 0.500
    • 0.068

  20. A student says two individual 95% confidence intervals for two means overlap, so the means cannot differ significantly. What is the correct evaluation?

    • Overlap of individual intervals is not a reliable test; the interval for the difference must be used
    • Overlap can only happen when the variances are unknown
    • Overlap means the difference in means is exactly zero
    • Overlap proves that the two means are equal

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