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 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 α%
- EITHER the raw data
- OR the sample mean , 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.
b)
The butcher claims the burgers weigh 300 grams. Comment on this claim with reference to the confidence interval.