Chapter 8, continued...

III. Interpretation of Confidence Intervals

Remember, we don’t know the population mean. We take a sample to estimate µ, then construct a confidence interval (CI) to provide some measure of accuracy for that estimate.

An accurate interpretation for a 95% CI:

“Before sampling, there is a 95% chance that the interval: will include µ.x

n196.

More interpretation.

In other words, if 100 samples are taken, each of size n, on average 95 of these intervals will contain µ.

Important: this statement can only be made before we sample, when x-bar is still an undetermined random variable. After we sample, x-bar is no longer a random variable, thus there is no probability.

An example of interpretation.

Suppose that the CJW company samples 100 customers and finds this month’s customer service mean is 82, with a population standard deviation of 20. We wish to construct a 95% confidence interval. Thus, =.05 and z.025=1.96.

Before vs. After sampling

• Before we sample, there is a 95% chance that µ will be in the interval:

• After sampling we create an interval:

82 ± 3.92, or (78.08 to 85.92).

We can only say that under repeated sampling, 95% of similarly constructed intervals would contain the true µ . This one particular interval may or may not contain µ .

xn

196.

IV. Interval Estimate of µ: Small Sample

A small sample is one in which n<30. If the population has a normal probability distribution, we can use the following methods. However, if you can’t assume the normal population, you must increase n30 so the Central Limit Theorem can be invoked.

A. The t-distribution

William Sealy Gosset (“student”) founded the t-distribution. An Oxford graduate in math and chemistry, he worked for Guinness Brewing in Dublin and developed a new small-sample theory of statistics while working on small-scale materials and temperature experiments. “The probable error of a mean” was published in 1908, but it wasn’t until 1925 when Sir Ronald A. Fisher called attention to it and its many applications.

The idea behind the t.

Specific t-distributions are associated with a different degree of freedom.

Degree of freedom: the # of observations allowed to vary in calculating a statistic = n-1.

As the degrees of freedom increase (n), the closer the t-distribution gets to the standard normal distribution.

B. An Example.

Suppose n=20 and you are constructing a 99% (=.01) confidence interval.

First we need to be able to read a t-table to find t.005. See Table 8.3 in the text.

x ts

n /2

The t-table.

0

/2

t/2

We need to find t.005 with 19 degrees of freedom in a t-table like Table 8.3.

I see whereyou’re going!

Our Example

D.F. .10 .05 .01 .025 .005....

18

19 1.328 1.729 2.093 2.539 2.861

How do I get back to the brewery?