1. STA 291 Summer 2008
Lecture 14
Dustin Lueker

2. Confidence Intervals
This interval will contain μ with a 100(1-α)% confidence
If we are estimating µ, then why it is unreasonable for us to know σ?
Thus we replace σ by s (sample standard deviation)
This formula is used for large sample size (n≥30)
If we have a sample size less than 30 a different distribution is used, the t-distribution, we will get to this later
STA 291 Summer 2008 Lecture 14

3. Interpreting Confidence Intervals
Incorrect statement
With 95% probability, the population mean will fall in the interval from 3.5 to 5.2
To avoid the misleading word “probability” we say
We are 95% confident that the true population mean will fall between 3.5 and 5.2
STA 291 Summer 2008 Lecture 14

4. Confidence Interval
Changing our confidence level will change our confidence interval
Increasing our confidence level will increase the length of the confidence interval
A confidence level of 100% would require a confidence interval of infinite length
Not informative
There is a tradeoff between length and accuracy
Ideally we would like a short interval with high accuracy (high confidence level)
STA 291 Summer 2008 Lecture 14

5. Facts about Confidence Intervals
The width of a confidence interval
Increases as the confidence level increases
Increases as the error probability decreases
Increases as the standard error increases
Increases as the sample size n decreases
Why are each of these true?
STA 291 Summer 2008 Lecture 14

6. Choice of Sample Size
Start with the confidence interval formula assuming that the population standard deviation is known
Mathematically we need to solve the above equation for n
STA 291 Summer 2008 Lecture 14

7. Example
About how large a sample would have been adequate if we merely needed to estimate the mean to within 0.5, with 95% confidence? Assume
Note: We will always round the sample size up to ensure that we get within the desired error bound.
STA 291 Summer 2008 Lecture 14

8. Confidence Interval for Unknown σ
To account for the extra variability of using a sample size of less than 30 the student’s t- distribution is used instead of the normal distribution
STA 291 Summer 2008 Lecture 14

9. t-distribution
t-distributions are bell-shaped and symmetric around zero
The smaller the degrees of freedom the more spread out the distribution is
t-distribution look much like normal distributions
In face, the limit of the t-distribution is a normal distribution as n gets larger
STA 291 Summer 2008 Lecture 14

10. Finding tα/2 Need to know α and degrees of freedom (df) α=.05, n=23
STA 291 Summer 2008 Lecture 14

11. Example
A sample of 12 individuals yields a mean of 5.4 and a variance of 16. Estimate the population mean with 98% confidence.
STA 291 Summer 2008 Lecture 14