Confidence Intervals
Notation: estimates vs estimators
- unknown parameter (a constant, i.e ) - estimate to : a number calculated based on data (i.e ) on RV that corresponds to (i.e is an outcome)
Estimates vs Estimators:
Estimates:
Estimators:
A 100%p confidence interval for
Example
Let
This does not mean that there is a 95% chance the interval contains the
Instead, it means 95% of constructed CIs fall around (i.e contain)
Interpretation of confidence intervals:
is nonsense mathematically. There is no randomness, either
Really what we are asking is
[!steps]
- Construct the Pivotal quantity
- Using the Pivotal Distribution, construct the Coverage Interval
- Estimate the Coverage Interval using your data
[!example]
Suppose that we have
Construct a 95% confidence interval for
solution
Construct the Pivotal Quantity
A RV which is function of
Using the Pivotal Distribution, construct the Coverage Interval
best estimate of
CI is:
is the confidence level
A confidence interval is a range of estimates for an unknown parameter, which is computed at a designated confidence level.
The statement "on average, a household lightbulb will last
When making a conclusion about the mean of a population, the confidence interval is calculated by:
where:
is the population mean (estimated by the sample mean) is the sample mean is the standard deviation is the confidence level z-score is the sample size
When making a conclusion about the proportion of a population, the confidence interval is calculated by:
where:
is the population proportion (estimated by the sample proportion) is the sample proportion (i.e the probability of success) is the probability of failure is the confidence level z-score is the sample size
This formula should be used when the sample is much smaller than the population
The confidence level z-scores can be obtained by working backwards from a z-score table. The most common ones are:
| Confidence Level | 90% | 95% | 99% |
|---|---|---|---|
| z | 1..65 | 1.96 | 2.58 |
A paint manufacturer knows that drying times for its latex paints have a standard derivation of 10.5 minutes. Twenty test areas of equal size are painted and the mean drying time is found to be 75.4 minutes. Calculate a 95% confidence interval for the actual mean drying time of the paint.
solution
We have:
using the equation:
so the confidence interval is
That is, the drying time of the paint is
In a recent election, the mayor got 72% of the vote, but only 5000 voters turned out. Construct a 90% confidence interval for the proportion of the people who support the mayor.
solution
We have:
using the equation:
so the confidence interval is 71% to 73% with a confidence level of 90%.