From the sampling distribution of X-bar to inference. (The first day of Chapter 14.)

Interval Estimates

Recall the example: Estimate the average height, in inches of male UT students.

We took a random sample of 49 male UT students, and the sample mean is 69.7 inches. So we estimate that the average height of all UT male students is approximately 69.7 inches.

But how close is "approximately" here? We'll indicate that by an interval, like 68.7 to 70.7 inches. (In this interval I used a "margin of error" of 1 inch, meaning that I went 1 inch on either side of the sample mean.)

And how confident do we want to be that we capture the population mean in our interval? Let's assume we want to be 95% confident.

As you might guess, the size of the interval and how confident we can be that it captures the population mean are related. They are related by the sampling distribution of the X-bar. See the discussion in the first three sections of Chapter 14.

Brief answer: We think of the sampling distribution of X-bar for samples of size 49. We determine how much variablility that distribution has, and we use that to see how far out into the tails we have to go to capture 95% of the area of the sampling distribution of X-bar. That tells us how wide to make our interval.
Answer: Two standard deviations.

We know (from other sources) that the the standard deviation of the heights of male UT students is 2.1 inches. When we have found that, and have decided on a 95% confidence level, we have all we need to answer this question.
Answer: We know that the standard deviation of the sampling distribution is 2.1 / sqrt(49) = 0.3. So the "margin of error" we need is really 2(0.3) = 0.6.
Then my 95% interval is 69.7 - 0.6 = 69.1" to 69.7+0.6 = 70.3"

Answer, with interpretation: I have 95% confidence that the mean height for the entire population of UT men is between 69.1" and 70.3".

In class, we also did the 68% confidence interval (margin of error of 1 standard deviation) and the 99.7% confidence interval (margin of error of 3 standard deviations.)

Then demonstrate finding the z-scores for the various confidence intervals: 90% and 99% (You will not have to actually carry this out. You can just copy the needed z-scores onto your notes for the test.)

Then read and discuss Example 14.3 about the healing of skin wounds.

Then do Exercise 14.5 b.

Then review the conditions on page 367. Notice that, because of the Central Limit Theorem, the condition about the population being normally distributed is not very important if the sample size is reasonably large - 30 or more.

Then do Exercise 14.5a.

 

Tests:

See exercise 14.7.

We will look at the claim of mu = 5 and describe the sampling distribution of X-bar for samples of size 6.

Then we look at the data we found and where its sample mean falls in that sampling distribution. If it is way out in the tail, then we decide the data is NOT consistent with the claim that mu = 5.

If the data is not way out in the tail, then we decide that the data IS consistent with the claim that mu = 5.

--------------- (I stopped here on Thursday.)

We can get numerical measures of all of this by computing a probability in that sampling distribution, and we'll call that the p-value of the data. Then we'll interpret it and use it to complete the analysis. (Small p-value means reject the null hypothesis. Not-so-small p-value means that we do not reject the null hypothesis.)

14.17. Here's where we actually compute those p-values.

See solutions to these two problems.

After finishing Chapter 14, you can review my overview of hypothesis testing.