Edexcel IAL - Statistics 3- 3.3 Confidence Intervals and Interpretation- Study notes  - New syllabus

Confidence Intervals

A confidence interval is a range of values that is used to estimate an unknown population parameter, such as the population mean. It is constructed from sample data and gives an indication of the reliability of the estimate.

Concept of a Confidence Interval 

A confidence interval for a population mean \( \mathrm{\mu} \) has the general form

\( \mathrm{estimate \pm margin\ of\ error} \)

For large samples or when the population standard deviation is known, a confidence interval for the mean is based on the sample mean \( \mathrm{\bar{X}} \).

Interpretation of a Confidence Interval

A confidence interval with confidence level \( \mathrm{(1-\alpha)\times100\%} \) means that:

If many random samples were taken and a confidence interval calculated from each, then approximately \( \mathrm{(1-\alpha)\times100\%} \) of those intervals would contain the true population parameter

It does not mean that there is a probability of \( \mathrm{(1-\alpha)} \) that the parameter lies in the particular interval calculated.

Confidence Interval for a Population Mean

If the population is normally distributed (or the sample size is sufficiently large), a confidence interval for the mean is given by

\( \mathrm{\bar{X} \pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}} \)

where:

\( \mathrm{z_{\alpha/2}} \) is the critical value from the standard normal distribution

\( \mathrm{\sigma/\sqrt{n}} \) is the standard error of the mean

Common confidence levels are 90%, 95%, and 99%.

Effect of Sample Size and Confidence Level

• Increasing the sample size reduces the width of the confidence interval
• Increasing the confidence level increases the width of the confidence interval

There is a trade-off between precision and confidence.

Link with Hypothesis Tests

Confidence intervals are closely related to hypothesis testing.

For a two-tailed hypothesis test at significance level \( \mathrm{\alpha} \):

The null hypothesis value of the parameter lies inside the \( \mathrm{(1-\alpha)\times100\%} \) confidence interval if and only if the null hypothesis is not rejected

The null hypothesis value lies outside the confidence interval if and only if the null hypothesis is rejected

Thus, confidence intervals provide an alternative way of presenting the result of a hypothesis test.

Key Points to Remember

• A confidence interval estimates an unknown population parameter
• The confidence level reflects the long-run success rate of the method
• Confidence intervals and hypothesis tests give consistent conclusions