Confidence Interval for the Mean (DP IB Maths: AI HL)

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Dan

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Dan

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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 is known σ²
    • On your GDC enter:
      • the standard deviation σ and the confidence level α%
      • EITHER the raw data
      • OR the sample mean x with bar on top 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 s subscript n minus 1 end subscript superscript 2
    • On your GDC enter:
      • the confidence level α%
      • EITHER the raw data
      • OR the sample mean x with bar on top, the value of sn-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 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

Worked example

Cara wants to check the mean weight of burgers sold by a butcher. The weights of the burgers are assumed to be normally distributed. Cara takes a random sample of 12 burgers and finds that the mean weight is 293 grams and the standard deviation of the sample is 5.5 grams.

a)
Find a 95% confidence interval for the population mean, giving your answer to 4 significant figures.

4-12-3-ib-ai-hl-confidence-intervals-a-we-solution

b)
The butcher claims the burgers weigh 300 grams. Comment on this claim with reference to the confidence interval.

4-12-3-ib-ai-hl-confidence-intervals-b-we-solution

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Dan

Author: Dan

Expertise: Maths

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.