Confidence intervals

Want to estimate parameters such as (population mean) or p (population proportion)

Obtain a SRS and use our estimators, and Even though these are good estimators, they will rarely be exactly on target.

For this reason, we typically include a margin of error along with the estimates, forming an interval.

Estimates and errors

x.p̂

Margins of error

Calculate margin of error based on info from the SRS and the sampling dist. of the estimator.

The margins of error need to be larger - you will have to allow for the possibility of greater error – if you:– Have a small sample– Want higher level of confidence in your interval– Have a large population variance

Confidence level

How large does the margin of error have to be so that you can be 100C % confident in an interval?

Can you be 100% confident that an interval encloses the parameter?– Only way is to give an interval including all possible

values the parameter can take, such as the interval (0,1) for proportions - completely useless!

– We have to settle for less than 100% confidence.

Confidence interval (CI) for

For large samples, CLT says that is approx. normally dist. with mean and std. deviation /sqrt(n).

Use normal dist. and std. deviation of to find margins of error: +/- z* /sqrt(n).

z* is the z-score that marks off top (1-C)/2 % of the standard normal curve

Level C CI: +/- z* /sqrt(n)

x

x

x

Example: mean age of customers

To determine the average age of its customers, a men’s clothing manufacturer took a SRS of 50 customers and found the average age of the sample was 36. If we know the standard deviation of the age of all the customers is 12:– What’s a 95% CI for the mean age of all customers?– Suppose you want to make the 95% CI narrower,

say +/- 2 years. How large a sample is required?

What do we mean by “confidence”?

Having 95% confidence in an interval is NOT same as saying that there is a 95% probability that the parameter lies in the interval!

Once the sample has been taken and used to calculate a CI, there is nothing random left- the interval either encloses the parameter or not.

We ARE saying that 95% of samples will yield CIs that enclose the parameter