Formula for a confidence interval for a Normal mean (\sigma known)?

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

Correct meaning of a 95\% confidence interval?

If many samples were taken and an interval found from each in the same way, about 95\% of the intervals would contain \mu.

Why is it wrong to say there is a 95% probability that \mu is in this interval?

Once calculated, the interval is fixed and \mu is a fixed number; the 95\% refers to the method over repeated samples.

How do you find the minimum n for a given maximum width w?

Solve 2z\frac{\sigma}{\sqrt n}\le w for n and round up.

What happens to the width when the confidence level rises?

It increases, because z is larger.

What does a 95\% confidence interval correspond to in a hypothesis test?

The values of \mu not rejected by a two-tailed test at the 5\% level.

A claimed mean lies outside the 99\% interval. Conclusion?

Reject H0 at the 1\% significance level (two-tailed): there is evidence the mean differs from the claim.

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

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What is a confidence interval?

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What is a confidence interval?: A range of values, calculated from a sample, that is likely to contain the unknown population parameter.
Formula for a confidence interval for a Normal mean ( $\sigma$ known)?: $\bar x\pm z\,\frac{\sigma}{\sqrt n}$
Standard error of the sample mean?: $\frac{\sigma}{\sqrt n}$
$z$ for a $90\%$ confidence interval?: $1.645$
$z$ for a $95\%$ confidence interval?: $1.96$
$z$ for a $99\%$ confidence interval?: $2.576$
Correct meaning of a $95\%$ confidence interval?: If many samples were taken and an interval found from each in the same way, about $95\%$ of the intervals would contain $\mu$ .
Why is it wrong to say there is a 95% probability that $\mu$ is in this interval?: Once calculated, the interval is fixed and $\mu$ is a fixed number; the $95\%$ refers to the method over repeated samples.
Width of the interval?: $2z\frac{\sigma}{\sqrt n}$
How do you find the minimum $n$ for a given maximum width $w$ ?: Solve $2z\frac{\sigma}{\sqrt n}\le w$ for $n$ and round up.
What happens to the width when the confidence level rises?: It increases, because $z$ is larger.
How is the width changed by quadrupling $n$ ?: It is halved.
What does a $95\%$ confidence interval correspond to in a hypothesis test?: The values of $\mu$ not rejected by a two-tailed test at the $5\%$ level.
A claimed mean lies outside the $99\%$ interval. Conclusion?: Reject $H_0$ at the $1\%$ significance level (two-tailed): there is evidence the mean differs from the claim.

Exam questions on Confidence intervals for a Normal mean

The mass of a packet of flour, in grams, is Normally distributed with standard deviation $6$ . A random sample of $25$ packets has mean mass $503.2$ g.
The label says the mean mass is $500$ g. Use your interval from part (b) to comment on this claim, explaining the link with a hypothesis test.2 marks
A random sample of $36$ observations is taken from a Normal distribution with standard deviation $2.4$ . The sample mean is $13.0$ .
The sample size is to be increased so that the width of the $90\%$ confidence interval is at most $0.8$ . Find the smallest sample size required.2 marks
The time of one machine cycle, in seconds, is Normally distributed with standard deviation $0.5$ . The mean of a random sample of $40$ cycles is $12.84$ s.
Calculate a $99\%$ confidence interval for the mean cycle 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).