Chapter 8, continued.... III. Interpretation of Confidence Intervals Remember, we don’t know the...

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Chapter 8, continued...

Transcript of Chapter 8, continued.... III. Interpretation of Confidence Intervals Remember, we don’t know the...

Page 1: Chapter 8, continued.... III. Interpretation of Confidence Intervals Remember, we don’t know the population mean. We take a sample to estimate µ, then.

Chapter 8, continued...

Page 2: Chapter 8, continued.... III. Interpretation of Confidence Intervals Remember, we don’t know the population mean. We take a sample to estimate µ, then.

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.

Page 3: Chapter 8, continued.... III. Interpretation of Confidence Intervals Remember, we don’t know the population mean. We take a sample to estimate µ, then.

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.

Page 4: Chapter 8, continued.... III. Interpretation of Confidence Intervals Remember, we don’t know the population mean. We take a sample to estimate µ, then.

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.

Page 5: Chapter 8, continued.... III. Interpretation of Confidence Intervals Remember, we don’t know the population mean. We take a sample to estimate µ, then.

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.

Page 6: Chapter 8, continued.... III. Interpretation of Confidence Intervals Remember, we don’t know the population mean. We take a sample to estimate µ, then.

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.

Page 7: Chapter 8, continued.... III. Interpretation of Confidence Intervals Remember, we don’t know the population mean. We take a sample to estimate µ, then.

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.

Page 8: Chapter 8, continued.... III. Interpretation of Confidence Intervals Remember, we don’t know the population mean. We take a sample to estimate µ, then.

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.

Page 9: Chapter 8, continued.... III. Interpretation of Confidence Intervals Remember, we don’t know the population mean. We take a sample to estimate µ, then.

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

Page 10: Chapter 8, continued.... III. Interpretation of Confidence Intervals Remember, we don’t know the population mean. We take a sample to estimate µ, then.

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!

Page 11: Chapter 8, continued.... III. Interpretation of Confidence Intervals Remember, we don’t know the population mean. We take a sample to estimate µ, then.

Our Example

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

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19 1.328 1.729 2.093 2.539 2.861

How do I get back to the brewery?