Class 5 Estimating  Confidence Intervals

Estimation of  Imagine that we do not know what  is, so we would like to estimate it. In order to get a point estimate of , we would take a sample and compute. Is there a better way? Should we use our sample to compute something else that would yield better guesses of  ? If you take a sample and (1) multiple the sample values by any amount that you like, and (2) add the results together, you can not do better than.

Estimation (cont.) For example, take a sample of size 4 from a normal population and compute where X i is the i th sample value. Then W has a normal distribution, E(W) = , but What does this mean about W?

Estimation (cont.) An estimator, Y, of  is said to be unbiased if E(Y) = . Thus, W and are unbiased. In fact, it can be shown that is the best linear unbiased estimator (BLUE) of  in the sense that, among all linear unbiased estimators, it has the smallest variance.

Building Interval Estimates: The Confidence Interval We do not know , but if we did (and if we had a large enough sample), we would know exactly how was distributed. This tells us where will probably fall. But we have a different problem: we see. Where does  probably fall? For a given probability, this is called a confidence interval.

The Confidence Interval Let z  be the point on the standard normal distribution that cuts off  % in the upper tail. A 100(1-  )% confidence interval for  when the normality of is justifiable, and  is known: What information comes from the sample?

There is a 1 -  probability that the value of a sample mean will provide a sampling error of or less. Sampling distribution of Sampling distribution of Probability Statements About the Sampling Error    /2 1 -  of all values 1 -  of all values

Example Incomes in a community are known to be normally distributed with  = $2000. In order to compute a 90% confidence interval for , you take a sample of 400 incomes and determine that = $24,000. What is z  /2 ? Then

Example (cont.) What is true about this interval? How would it change if we contructed a 95% confidence interval?

Example (cont.) Now assume that you wish to construct a 95% confidence interval using a sample of If = $24,000, then our interval is (23,902, 24,098). We would like our interval to be small. What are the two things we can change?

Confidence Intervals for  --  Unknown If you did not know , what would you use to estimate  ? To construct a 100(1-  )% Confidence Interval for  when  is unknown, compute: Where t  /2,n-1 is the value on the t distribution with n-1 degrees of freedom that cuts off  /2 of the distribution in the upper tail.

t distributions The t distribution is a family of similar probability distributions. A specific t distribution depends on a parameter known as the degrees of freedom. As the number of degrees of freedom increases, the difference between the t distribution and the standard normal probability distribution becomes smaller and smaller. A t distribution with more degrees of freedom has less dispersion. The mean of the t distribution is zero.

t distributions t Value: Assume a sample of size 10. At 95% confidence, 1 -  =.95,  =.05, and  /2 =.025. t.025,9 is based on n - 1 = = 9 degrees of freedom. In the t distribution table we see that t.025,9 =

t distributions (Once again) A t distribution is mound shaped and symmetric about 0. There is a different t distribution for every different degree of freedom. The t distribution approaches the z (standard normal) distribution as n . t.1,20 = t.05,10 = t.025,  = t.025,30 =

Example Incomes in a community are normally distributed. A sample of size 4 is taken, and the following incomes are found: Estimate the mean income of the community with a 95% confidence interval. 21,000 24,500 22,500 22,000 Let’s begin by doing this in EXCEL.

Our example in EXCEL Input the data into an EXCEL column. Select tools/data analysis/descriptive statistics. Input data range and output range. Select summary statistics and confidence level for mean. How was the standard error computed? What is the ratio of the confidence level and the standard error?

Summary