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

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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 standard error used in a confidence interval for a single mean $\mu$ with known $\sigma$?

    • $\sigma / n$
    • $\sqrt{\sigma / n}$
    • $\sigma / \sqrt{n}$
    • $\sigma^2 / n$

  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 standard error of $\bar{X} - \bar{Y}$ for independent samples with known population variances?

    • $\sqrt{(\sigma_x^2 + \sigma_y^2)/(n_x + n_y)}$
    • $\sigma_x/\sqrt{n_x} + \sigma_y/\sqrt{n_y}$
    • $\sqrt{\sigma_x^2/n_x + \sigma_y^2/n_y}$
    • $\sigma_x^2/n_x + \sigma_y^2/n_y$

  4. Under $H_0$, what is the distribution of the test statistic for testing $\mu_X - \mu_Y$ with known variances?

    • $\chi^2_1$
    • $F_{n_1, n_2}$
    • $N(0, 1)$
    • $t_{n-1}$

  5. Which assumption allows $z$-tests for sample means when populations are non-normal?

    • Equal sample sizes
    • Zero population mean
    • Small sample size
    • Large sample size

  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. Increasing the confidence level from 95% to 99% has what effect on a confidence interval?

    • Halves its width
    • Makes it wider
    • Has no effect
    • Makes it 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. A test statistic is $z = 4.60$. At the 5% two-tailed level (critical value 1.96), what is the conclusion?

    • Inconclusive
    • Accept $H_0$
    • Reject $H_0$
    • Reject $H_1$

  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 $(1.2, 6.8)$. Is $H_0: \mu_X = \mu_Y$ rejected at 5% significance?

    • No, includes zero
    • No, excludes zero
    • Yes, includes zero
    • Yes, excludes zero

  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). Is H0: mu_x = mu_y rejected at 5%?

    • Always
    • No
    • Only at 1%
    • Yes

  18. Which distribution does the standardized sample mean follow when the population is normal and sigma is known?

    • F distribution
    • Student t
    • Chi-squared
    • Standard normal

  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. Which confidence interval best tests whether two population means are significantly different?

    • Difference of means
    • Pooled proportion
    • Combined variance
    • Individual sample means

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