Further Mathematics · Pearson Edexcel 9FM0 · Option FS2 · Topic 7

Confidence intervals and tests using the t-distribution

The t-distribution for small samples: confidence intervals and tests for means, including paired data.

3 sub-topics
Taught in Year 13
48 practice questions
3 animated lessons

Practice questions (48) Lessons Worksheets All Edexcel Further Mathematics topics

What the specification says

Every sub-topic Pearson Edexcel 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.

7.1t-tests and intervals for a normal mean (variance unknown)Y13
7.2Paired t-testY13
7.3Two-sample t-test (equal unknown variances)Y13

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Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 48 questions in this topic, free, no account needed.

Confidence intervals and tests using the t-distribution

Not needed for Year 12Easy

Choosing between z and t · Year 13 · 1 mark

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

Total1

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Confidence intervals and tests using the t-distribution

Not needed for Year 12Medium

A t statistic · Year 13 · 1 mark

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

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$t$ ${}= \dfrac{\bar{x} - \mu_0}{s/\sqrt{n}}$ , compared with $t_{15}$ .A: 2

Total1

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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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.

(a)
Test, at the 5% level, whether the population mean differs from 7.5.
(5)
(b)
State an assumption you have made.
(1)

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Part (a)5 marks

$H_0: \mu$ ${}= 7.5, \ H_1: \mu \neq 7.5$
$\sigma$ is unknown, so use the $t$ distribution. $t$ ${}= \frac{7.34 - 7.5}{0.38/\sqrt{10}}$ ${}= -1.331$
Two-tailed, ν = n − 1. $t_{9}(2.5\%) = 2.262$
Do not reject H₀: there is insufficient evidence that the mean differs from 7.5.

Part (b)1 mark

The population is normally distributed.

Total6

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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