Further Mathematics · AQA 7367 · Option S · Topic SG

Inference: one-sample t-distribution

The t-test for a mean when the variance is unknown.

  • 1 sub-topic
  • Taught in Year 13
  • 17 practice questions
  • 1 animated lesson
Practice questions (17) Lessons Worksheets All AQA Further Mathematics topics

What the specification says

Every sub-topic AQA can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

  1. SG1t-test for the mean of a normal distribution with unknown varianceY13

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Confidence intervals and tests using the t-distribution

Not needed for Year 12Easy

Choosing between z and t · Year 13 · 1 mark

  1. A small sample is taken from a Normal population. When should the t-distribution be used rather than the Normal distribution?
    (1)

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  1. The extra uncertainty from estimating the variance makes the tails heavier.CC: When the population variance is unknown and is estimated from the sample

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Confidence intervals and tests using the t-distribution

Not needed for Year 12Medium

A t statistic · Year 13 · 1 mark

  1. A sample of size 16 from a Normal population has mean 52 and unbiased standard deviation estimate 4. To test H0:μ=50H_0: \mu = 50, what is the value of the test statistic?
    (1)

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  1. tt=xˉ−μ0s/n{}= \dfrac{\bar{x} - \mu_0}{s/\sqrt{n}}, compared with t15t_{15}.A: 2

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Confidence intervals and tests using the t-distribution

Not needed for Year 12Hard

A t-test for a normal mean · Year 13 · 6 marks

A random sample of 10 observations from a normal distribution has mean 7.34 and unbiased standard deviation estimate 0.38.
  1. (a)
    Test, at the 5% level, whether the population mean differs from 7.5.
    (5)
  2. (b)
    State an assumption you have made.
    (1)
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Part (a)5 marks

  1. H0:μH_0: \mu=7.5, H1:μ≠7.5{}= 7.5, \ H_1: \mu \neq 7.5
  2. σ\sigma is unknown, so use the tt distribution.tt=7.34−7.50.38/10{}= \frac{7.34 - 7.5}{0.38/\sqrt{10}}=−1.331{}= -1.331
  3. Two-tailed, ν = n − 1.t9(2.5%)=2.262t_{9}(2.5\%) = 2.262
  4. Do not reject H₀: there is insufficient evidence that the mean differs from 7.5.

Part (b)1 mark

  1. The population is normally distributed.

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