What does a 95\% confidence level mean?

About 95\% of intervals built in this way from repeated random samples would contain the true parameter.

Confidence limits for a Normal mean with \sigma known?

\bar x\pm z\frac{\sigma}{\sqrt n}.

How does a larger sample affect the interval?

It makes it narrower, since the standard error \frac{\sigma}{\sqrt n} is smaller.

How does a higher confidence level affect the interval?

It makes it wider, because z is larger.

How do you find the sample size needed for a given width w?

n\geq\left(\frac{2z\sigma}{w}\right)^2, rounded up to a whole number.

How do you find \bar x from a symmetrical confidence interval?

It is the midpoint of the lower and upper limits.

Is a 95\% interval a range containing 95\% of the population values?

No. It is an interval for the mean, not a range for individual values.

How is a 95\% interval linked to a hypothesis test?

If \mu0 lies outside it, reject H0:\mu=\mu0 in a two-tailed test at the 5\% level.

If \mu0 lies inside the interval, what do you conclude?

Do not reject H0: there is insufficient evidence at the corresponding level that \mu\neq\mu0.

Should the sample size be rounded up or down?

Up, so that the interval is no wider than required.

Confidence intervals for a Normal meanEdexcel International A Level Further Maths: Flashcards

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What is a confidence interval?: A range of values, calculated from a sample, that is likely to contain the population parameter.
What does a $95\%$ confidence level mean?: About $95\%$ of intervals built in this way from repeated random samples would contain the true parameter.
Confidence limits for a Normal mean with $\sigma$ known?: $\bar x\pm z\frac{\sigma}{\sqrt n}$ .
$z$ values for $90\%$ , $95\%$ and $99\%$ intervals?: $1.645$ , $1.96$ and $2.576$ .
What is the standard error of $\bar X$ ?: $\frac{\sigma}{\sqrt n}$ .
Width of a confidence interval for a mean?: $2z\frac{\sigma}{\sqrt n}$ .
How does a larger sample affect the interval?: It makes it narrower, since the standard error $\frac{\sigma}{\sqrt n}$ is smaller.
How does a higher confidence level affect the interval?: It makes it wider, because $z$ is larger.
How do you find the sample size needed for a given width $w$ ?: $n\geq\left(\frac{2z\sigma}{w}\right)^2$ , rounded up to a whole number.
How do you find $\bar x$ from a symmetrical confidence interval?: It is the midpoint of the lower and upper limits.
Is a $95\%$ interval a range containing $95\%$ of the population values?: No. It is an interval for the mean, not a range for individual values.
How is a $95\%$ interval linked to a hypothesis test?: If $\mu_0$ lies outside it, reject $H_0:\mu=\mu_0$ in a two-tailed test at the $5\%$ level.
If $\mu_0$ lies inside the interval, what do you conclude?: Do not reject $H_0$ : there is insufficient evidence at the corresponding level that $\mu\neq\mu_0$ .
Should the sample size be rounded up or down?: Up, so that the interval is no wider than required.

Exam questions on Confidence intervals for a Normal mean

The masses of bags of flour are Normally distributed with a known standard deviation of $8$ g. A random sample of $25$ bags has a mean mass of $1012$ g.
Find a $90\%$ confidence interval for the mean mass of a bag.2 marks
A $95\%$ confidence interval for the mean $\mu$ of a Normal population with known standard deviation $10$ is $(49.2,\ 54.8)$ . It was calculated from a random sample.
Use the confidence interval to carry out a test of $H_0:\mu=50$ against $H_1:\mu\neq50$ at the $5\%$ significance level, stating your conclusion.2 marks
The time taken, in minutes, for a train journey is Normally distributed with a known standard deviation of $4.5$ minutes. A random sample of $36$ journeys has a mean time of $52.4$ minutes.
Find a $99\%$ confidence interval for the mean journey time.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).