SH3-SH4 Making inferences and small-sample intervals Quiz: AQA Further Maths, Unit 4

The 20 questions

1. When a normal population has unknown variance and the sample is small, which distribution is used to construct a symmetric confidence interval for the mean?

   • The chi-squared distribution with n - 1 degrees of freedom
   • The t-distribution with n degrees of freedom
   • The t-distribution with n - 1 degrees of freedom
   • The standard normal distribution with n degrees of freedom

2. Which expression gives the symmetric confidence interval for the mean of a normal population with unknown variance, from a small sample of size n with mean xbar and standard deviation s?

   • xbar ± t(n-1) × s × sqrt(n)
   • xbar ± t(n-1) × s^2/n
   • xbar ± t(n-1) × s/sqrt(n)
   • xbar ± z × s/sqrt(n)

3. For a 95% symmetric confidence interval based on a sample of size 10, which t value should be used?

   • 2.093
   • 1.960
   • 2.228
   • 2.262

4. A sample of n = 10 from a normal population has mean 20 and standard deviation 3. What is the 95% confidence interval for the population mean, to 2 decimal places?

   • (17.85, 22.15)
   • (18.14, 21.86)
   • (19.05, 20.95)
   • (16.20, 23.80)

5. A 95% symmetric confidence interval for a mean, from a sample of size 10, is (17.85, 22.15). Which conclusion is correct?

   • The value 18 is the lower limit of a one-sided interval, so the interval cannot be used
   • The interval is invalid because the sample size of 10 is too small to give any conclusion
   • The value 18 lies inside the interval, so 18 is a plausible value for the mean at the 5% significance level
   • The value 18 lies outside the interval, so the mean differs significantly from 18 at the 5% level

6. Two independent 95% confidence intervals for two population means are (10.2, 14.8) and (13.5, 18.1). Which statement is correct?

   • Overlapping intervals do not by themselves prove the two population means are equal, so a formal comparison is needed
   • The overlap proves that the population means are equal at the 95% level
   • The overlap shows that the two sample means must lie within 1.3 of each other
   • Because the intervals overlap, the difference between the means is definitely not significant at the 95% level

7. A sample of n = 5 from a normal population with unknown variance has standard deviation s = 2. Using t = 2.776, what is the margin of error for a 95% confidence interval, to 2 decimal places?

   • 2.48
   • 1.75
   • 2.78
   • 1.96

8. A sample of n = 5 from a normal population with unknown variance has standard deviation s = 2. Which margin of error gives a 99% confidence interval for the mean, to 2 decimal places?

   • 4.12, using t = 4.604 with 4 degrees of freedom
   • 2.30, using z = 2.576
   • 2.48, using t = 2.776
   • 1.79, using z = 2.0

9. Why is the t-distribution, rather than the standard normal distribution, used for a small-sample interval when sigma is unknown?

   • Small samples always come from discrete populations, which require the t-distribution
   • The sample mean is not normally distributed for any sample size when sigma is unknown
   • Estimating sigma by s adds uncertainty, so the standardised mean has heavier tails than the normal distribution
   • t critical values are always smaller than z values, which gives narrower intervals for small samples

10. How does the t-distribution change as its degrees of freedom increase?

    • It becomes skewed to the right, away from symmetry
    • It approaches the standard normal distribution
    • It becomes identical to the chi-squared distribution with the same degrees of freedom
    • It becomes more spread out, with heavier tails

11. Two samples have the same mean 20 and the same standard deviation 3, one with n = 10 and one with n = 30. Which 95% confidence interval is wider, and why?

    • The n = 30 interval, because the t value increases with the sample size
    • The n = 10 interval, because both its t value and its standard error are larger
    • The n = 30 interval, because a larger sample has a larger variance
    • They have the same width, because the sample standard deviations are equal

12. A sample of n = 20 from a normal population with unknown variance has mean 45 and standard deviation 6. What is the margin of error for a symmetric 95% confidence interval for the mean, to 2 decimal places?

    • 2.57
    • 3.20
    • 2.63
    • 2.81

13. A 95% symmetric confidence interval for a population mean, based on n = 15 normal observations, is (33.2, 40.8). What are the sample mean and sample standard deviation, to 2 decimal places?

    • Mean 37.0 and standard deviation 6.86
    • Mean 36.0 and standard deviation 6.86
    • Mean 37.0 and standard deviation 1.77
    • Mean 37.0 and standard deviation 3.80

14. A 95% confidence interval for a population mean is (-0.4, 2.2). Which statement is correct?

    • Yes, because the midpoint of 0.9 is closer to 0 than to any other value
    • No, because the interval is not symmetric about 0, so the mean cannot be 0
    • No, because 0 lies inside the interval, so the mean cannot be 0
    • Yes, 0 lies inside the interval, so the hypothesis that the mean is 0 is not rejected at the 5% significance level

15. A symmetric 99% confidence interval for the mean of a normal population with known variance is (48.2, 53.8), from the same sample. What is the corresponding 95% confidence interval?

    • (47.15, 54.85)
    • (48.87, 53.13)
    • (49.20, 52.80)
    • (48.20, 53.80)

16. A sample of n = 8 from a normal population with unknown variance has standard deviation s = 2.5. Using t = 2.365 with 7 degrees of freedom, what is the margin of error for a 95% interval for the mean, to 2 decimal places?

    • 2.21
    • 1.73
    • 2.37
    • 2.09

17. Why can a 95% t-interval based on a small sample be misleading when the population is strongly skewed?

    • Skew makes the sample mean equal to the sample median, so the interval collapses to a single point
    • The normality assumption behind the t-interval may fail, so the stated 95% confidence level may not be accurate
    • The t-interval is only valid for discrete populations, so skewed continuous data cannot be used
    • Skew forces the use of the chi-squared distribution in place of the t-distribution

18. A 95% symmetric confidence interval for a mean is computed as xbar ± 2.262 × s/sqrt(10), with s = 3. If xbar = 20, what is the upper limit, to 2 decimal places?

    • 21.86
    • 22.26
    • 26.79
    • 22.15

19. A 95% confidence interval for a mean uses n = 10 observations (width 4.29). If the sample size increases to n = 40 with the same s = 3 and the same mean 20, the 95% t-interval has t = 2.023 with 39 degrees of freedom. What is its approximate width?

    • About 4.29
    • About 1.00
    • About 2.14
    • About 1.92

20. Which statement about interpreting confidence intervals is correct?

    • A narrower interval means the confidence level has increased for the same data
    • A narrower interval from a larger sample gives more precise information about the mean, without changing the confidence level
    • A higher confidence level always produces a more precise interval for the same data
    • A narrower interval proves that the population mean is closer to the sample mean

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