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?for 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?is?known?σ2
  • On your GDC enter:
    • the standard deviation?σ?and the confidence level?α%
    • EITHER the raw data

• 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 with?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