Test statistic for a one-sample t test?

t=\frac{\bar x-\mu0}{s/\sqrt n}

How does the t distribution compare with the Normal distribution?

Symmetric but with heavier tails; it approaches N(0,1) as n increases.

Assumption for a one-sample t test?

The population is Normally distributed.

Confidence interval for a mean with \sigma unknown?

\bar x\pm t{n-1}\frac{s}{\sqrt n}

Which t point gives a 95% two-tailed interval?

The upper 2.5\% point of t{n-1}.

Why is a t interval wider than a z interval?

It allows for the extra uncertainty from estimating \sigma by s.

When is a paired test used?

When observations come in linked pairs, e.g. the same subjects before and after.

What is the first step in a paired t-test?

Calculate the difference for each pair.

Test statistic for the paired t-test?

t=\frac{\bar d}{sd/\sqrt n} with n-1 degrees of freedom, n the number of pairs.

Why not use a two-sample test on paired data?

The samples are not independent; a paired test removes variation between individuals.

What should a conclusion say?

In context, as evidence at a stated level, e.g. 'there is evidence at the 5% level that the mean ...'.

t tests for a mean and the paired t-testEdexcel A-Level Further Maths: Flashcards

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When is a $t$ test used instead of a $z$ test for a mean?

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When is a $t$ test used instead of a $z$ test for a mean?: When the population variance is unknown and estimated by $s^2$ from the sample, with a Normal population.
Test statistic for a one-sample $t$ test?: $t=\frac{\bar x-\mu_0}{s/\sqrt n}$
Degrees of freedom for a one-sample $t$ test?: $n-1$
How does the $t$ distribution compare with the Normal distribution?: Symmetric but with heavier tails; it approaches $\mathrm{N}(0,1)$ as $n$ increases.
Assumption for a one-sample $t$ test?: The population is Normally distributed.
Confidence interval for a mean with $\sigma$ unknown?: $\bar x\pm t_{n-1}\frac{s}{\sqrt n}$
Which $t$ point gives a 95% two-tailed interval?: The upper $2.5\%$ point of $t_{n-1}$ .
Why is a $t$ interval wider than a $z$ interval?: It allows for the extra uncertainty from estimating $\sigma$ by $s$ .
When is a paired test used?: When observations come in linked pairs, e.g. the same subjects before and after.
What is the first step in a paired $t$ -test?: Calculate the difference for each pair.
Test statistic for the paired $t$ -test?: $t=\frac{\bar d}{s_d/\sqrt n}$ with $n-1$ degrees of freedom, $n$ the number of pairs.
What is assumed Normal in the paired test?: The population of differences.
Why not use a two-sample test on paired data?: The samples are not independent; a paired test removes variation between individuals.
What should a conclusion say?: In context, as evidence at a stated level, e.g. 'there is evidence at the 5% level that the mean ...'.

Exam questions on t tests for a mean and the paired t-test

The mass of a chocolate bar is Normally distributed. The label states a mean mass of $50$ g. A random sample of $10$ bars has sample mean $49.2$ g and sample standard deviation $s=2.1$ g. A test is carried out to see whether the mean mass is less than the label states.
The lower $5\%$ point of $t_9$ is $-1.833$ . Complete the test at the $5\%$ significance level and state your conclusion in context.2 marks
A random sample of $8$ observations from a Normal population has sample mean $24.3$ and sample standard deviation $s=1.9$ . The population variance is unknown.
Explain why the $t$ distribution is used rather than the Normal distribution, and state the assumption needed about the population.2 marks
A supplier claims that the mean length of its rods is $12.0$ cm. The lengths are Normally distributed. A random sample of $8$ rods has $\sum x=99.4$ and $\sum x^2=1236.48$ , where $x$ is the length in cm.
Calculate the sample mean and an unbiased estimate of the population variance.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).