When is a t-test used instead of a z-test?

When the population is normal, the variance is unknown and the sample is small, so s replaces \sigma.

How does the t-distribution differ from the normal?

It is symmetric but has heavier tails; it tends to the normal as \nu increases.

Formula for the unbiased estimate of variance?

s^2=\frac{1}{n-1}\left(\sum x^2-\frac{(\sum x)^2}{n}\right)

What assumption is needed for the t-test?

The population is normally distributed (and the sample is random).

What is the critical region?

The values of t beyond the critical value(s), for which H0 is rejected.

When is a one-tailed H1 justified?

When the question gives the direction before the data are seen.

If t is in the critical region, what do you do?

Reject H0 and state the conclusion in context, e.g. 'evidence that the mean is lower'.

If t is not in the critical region, what do you say?

Do not reject H0: insufficient evidence at that level, not proof that H0 is true.

Given a variance of 144, what is s?

s=\sqrt{144}=12, then use \frac{s}{\sqrt n}.

One-sample t-testAQA A-Level Further Maths: Flashcards

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What is the test statistic for a one-sample $t$-test?

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What is the test statistic for a one-sample $t$ -test?: $t=\frac{\bar x-\mu_0}{s/\sqrt n}$
How many degrees of freedom does a one-sample $t$ -test have?: $\nu=n-1$
When is a $t$ -test used instead of a $z$ -test?: When the population is normal, the variance is unknown and the sample is small, so $s$ replaces $\sigma$ .
How does the $t$ -distribution differ from the normal?: It is symmetric but has heavier tails; it tends to the normal as $\nu$ increases.
Formula for the unbiased estimate of variance?: $s^2=\frac{1}{n-1}\left(\sum x^2-\frac{(\sum x)^2}{n}\right)$
What assumption is needed for the $t$ -test?: The population is normally distributed (and the sample is random).
Which table column is used for a two-tailed $5\%$ test?: The $2.5\%$ one-tail column.
What is the critical region?: The values of $t$ beyond the critical value(s), for which $H_0$ is rejected.
What is $H_1$ for a two-tailed test?: $H_1:\mu\neq\mu_0$
When is a one-tailed $H_1$ justified?: When the question gives the direction before the data are seen.
If $t$ is in the critical region, what do you do?: Reject $H_0$ and state the conclusion in context, e.g. 'evidence that the mean is lower'.
If $t$ is not in the critical region, what do you say?: Do not reject $H_0$ : insufficient evidence at that level, not proof that $H_0$ is true.
Given a variance of $144$ , what is $s$ ?: $s=\sqrt{144}=12$ , then use $\frac{s}{\sqrt n}$ .

Exam questions on One-sample t-test

A café claims that the mean volume of its large coffees is $250$ ml. Volumes are normally distributed. A customer measures $9$ randomly chosen large coffees and finds a sample mean of $243.4$ ml and an unbiased estimate of the population variance of $81$ ml $^2$ .
A test is carried out at the $5\%$ significance level of $H_0:\mu=250$ against $H_1:\mu<250$ . The critical value of $t$ is $-1.860$ . State the conclusion of the test, in context.2 marks
A gardener claims that a variety of sunflower grows to a mean height of $180$ cm. Heights are normally distributed. A random sample of $10$ plants has heights $x$ cm with $\sum x=1840$ and $\sum x^2=339\,136$ .
A test is carried out at the $5\%$ significance level of $H_0:\mu=180$ against $H_1:\mu\neq180$ . The critical values of $t$ are $\pm2.262$ . State the conclusion of the test, in context.2 marks
An environment agency states that the mean nitrate concentration in a river is $25$ mg per litre. Concentrations are normally distributed. Residents suspect that the mean is higher, and the agency takes $6$ random readings, in mg per litre: $27.4,\ 25.9,\ 28.1,\ 26.5,\ 29.0,\ 27.7$ .
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).