Tarsia puzzle

in groups of 3-4

Confidence intervals

To calculate confidence intervals.To understand what a confidence

interval is.

Using the normal distribution curve below complete the following sentences:About …… of the population is within one standard deviation of the mean About …… of the population is within two standard deviations of the meanAbout …… of the population is within three standard deviations of the mean

95% confidence interval

Check out using your calculator or tables you agree with the 1.96 value

A 95% confidence interval for the population mean is the sample mean plus or minus 1.96 standard deviations

Summary

A confidence interval for the population mean, is constructed using the formula:

sample mean ± z multiplied by σ/√nwhere σ is the population standard deviation, n is the sample sizeand z is dependent on the confidence level.Eg z = 1.96 for a confidence level of 95%

A 95% confidence interval for the population mean is the sample mean plus or minus 1.96 x σ/√n

Example page 141 q8Pencils produced on a certain machine have lengths, in millimetres, which are normally distributed with a mean of μ and a standard deviation of 3. A random sample of 16 pencils are taken and the length x mm recorded for each one giving ∑x = 2848(a) State why X bar is normally distributed (b) Construct a 99% confidence interval for the mean

a) X has normal distributionb) (176.1,179.9)mm

Practice qs: Ex 5D page 134 qs 1,2,3

June 06 q4

The meaning of a confidence interval

It is important that you understand what is meant by a confidence interval. For example, if you find a 90% confidence interval, this does NOT mean that there is a 90% chance that the mean lies within the confidence interval. It means that if you take many samples of the same size and construct a 90% confidence interval from each one, then 90% of these intervals will contain the true population mean.

This may seem like a subtle distinction, or just a different wording. The important point, however, is that the true population mean is not random. Even though you don’t know what it is, it is fixed. Either a particular confidence interval contains the mean, or it doesn’t. So you cannot say that there is a 90% chance that a particular confidence interval contains the mean.

However, the confidence interval is a random variable as it is based on a random sample. So you can say that 90% of intervals calculated from samples of a given size will contain the true mean.

June 07 q3