Confidence Interval for the Mean (DP IB Applications & Interpretation (AI)): Revision Note

Confidence interval for μ

What is a confidence interval?

• It is impossible to find the exact value of the population mean when taking a sample

  • The mean of a sample is called a point estimate

• The best we can do is find an interval in which the exact value is likely to lie

  • This is called the confidence interval for the mean

• The confidence level of a confidence interval is the probability that the interval contains the population mean

  • Be careful with the wording!

    • The population mean is a fixed value so it does not make sense to talk about the probability that it lies within an interval

    • Instead we talk about the probability of an interval containing the mean

• Suppose samples were collected and a 95% confidence interval for the population mean was constructed for each sample

  • Then for every 100 intervals, we would expect on average 95 of them to contain the mean

    • 95 out of 100 is not guaranteed – it is possible that all of them could contain the mean

    • It is also possible (though very unlikely) that none of them contains the mean

How do I find a confidence interval for the population mean (μ)?

• You will be given data using a sample from a population

  • The population will be normally distributed

    • If not then the sample size should be large enough so you can use the Central Limit Theorem

• You will use the interval functions on your calculator

• Use a z-interval if the population variance σ² is known

  • On your GDC enter:

    • the standard deviation σ and the confidence level %

    • EITHER the raw data

    • OR the sample mean  and the sample size n

• Use a t-interval if the population variance is unknown

  • In this case the test uses the unbiased estimate for the variance 

  • On your GDC enter:

    • the confidence level %

    • EITHER the raw data

    • OR the sample mean , the value of s_n-1 and the sample size n

• Your GDC will give you the lower and upper bounds of the interval

  • It can be written as a < μ < b

What affects the width of a confidence interval?

• The width of a confidence interval is the range of the values in the interval

• The confidence level affects the width

  • Increasing the confidence level will increase the width

  • Decreasing the confidence level will decrease the width

• The size of the sample affects the width

  • Increasing the sample size will decrease the width

  • Decreasing the sample size will increase the width

How can I interpret a confidence interval?

• After you have found a confidence interval for μ you might be expected to comment on the claim for a value of μ

• If the claimed value is within the confidence interval then there is not enough evidence to reject the claim

  • Therefore the claim is supported

• If the claimed value is outside the interval then there is sufficient evidence to reject the claim

  • The value is unlikely to be correct