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13.02.2014
This chapter, like all the others, has just two ideas in it; the second one builds on the first. Actually, a sampling distribution is only hard to understand when you're reading the definition. The Definition: A sampling distribution is a frequency distribution of values of a statistic computed for all possible samples of size N drawn from a population, sampling with replacement. The way we usually sample is without replacement; you draw out a case and put it in your sample, then draw out another, etc. You'll draw all possible samples of a given size from the population; sample size is (as usual) N.
Once you've drawn all these samples, you compute some statistic, often the mean, for each sample. Now we've exhausted all the possible things that could happen when one case drawn is 100; the other one could be 100 (again), 105, 110, or 115. Well, if you draw 105 first, then put the case back in and draw again, you might draw out 100 next.
When you start figuring out combinations with 110, remember that you've already had the combination 100 and 110, as well as 105 and 110. Characteristics of Sampling Distributions: So now, having done all this work, I want you to begin noticing some things. Mean: Another thing you might notice just by eyeballing it, although it also works out mathematically if you want to take the time, is that the mean of the sampling distribution of the mean is the same as the mean of the original population. Standard Deviation: You can tell at a glance that the means of all the samples you drew are not the same. AND, the good news is that we don't have to go through all the usual steps from Chapter 3 to find this number. Now you'll notice by looking at the formula that the standard error of the mean gets larger when the population standard deviation is larger. Since variability in the sampling distribution is an indication of how much sampling error there is (because it is caused by sampling error), the smaller it is, the better. Central Limit Theorem: These three characteristics of sampling distributions--the normal shape, the mean equal to the population mean, and the procedure for computing their standard deviations--are all gathered into a rule called the central limit theorem. Sampling Distribution of the Proportion: A proportion is simply the decimal value of the fraction of cases in a particular category.
Let's think about proportions where there are only two categories into which cases can fit, for example, married and unmarried, men and women, drivers and non-drivers.
Working with Proportions: For the record, p represents the proportion of cases assigned a value of 1.
This does not mean that the pollsters are unsure exactly what per cent of the 2000 people are "in favor" or "opposed." Those numbers are reported correctly.


Confidence Intervals: So a confidence interval is a mathematical estimate of the likelihood and degree to which you're wrong when you generalize from a sample to a population. The confidence interval is a range constructed around your sample mean, which has a known probability of containing the actual population mean. Interval Estimation: Part of being honest with statistics is to admit you could be wrong, then to estimate the likelihood you're wrong, and to estimate how far wrong you most likely are. This blue area would include the means of 95% of the possible samples you could draw from the population. So, if we build a confidence interval around our particular sample mean that is 1.96 standard errors wide on either side of it, we have a 95% chance that that confidence interval will include the population mean.
Say you've used the College Readiness Inventory to test the college readiness of 100 high school seniors. So to compute standard error of the mean, we also need to know the population standard deviation, which is given as 7, and the size of N, which is given as 100. Even though we don't write the confidence interval as a range of values, it's good to keep in mind just what this means. The 90% Confidence Interval for the Population Mean: You can figure out how to construct a 90% confidence interval in the very same way you figured out the 95% interval.
So we figure out how big 1.65 standard errors are, and we have the interval size all worked out.
The 99% Confidence Interval for the Population Mean: Now try on your own to figure out the 99% confidence interval for our CRI example. Other Levels of Confidence: The most frequently used levels of confidence are the three for which you've just found z-scores.
Try this example: On the Physicians' Compassion Index, a sample of 62 general practitioners achieved a mean score of 79, with a standard deviation of 8. First we need to estimate the standard error of the mean; this will be an estimate because we don't know the population standard deviation, so will have to use an estimated population figure. Remember that standard deviations cannot be corrected directly; you must find the variance, correct that, then find the corrected standard deviation from that. There is a formula for correcting variance, which will be provided to you when you take the test.
Interval Estimation When N is Small: We've been operating under the assumption that our sampling distribution is a normal distribution. To get 95% of the area under this curve, we're going to have to go out from the mean further than 1.96 standard errors--we need a bigger z-score. The other piece of information you need to enter this table is degrees of freedom (df), so you know which row of the table to read.
So to find your z-score, you enter the table at the proper significance level and degrees of freedom, follow the correct row and column to the point at which they intersect, and use as a z-score the number you find at the intersection.


Note that the confidence interval is wider, as it should be with a flatter sampling distribution.
Interval Estimation When N is Small AND the Population Standard Deviation is Unknown: This can happen, and it involves remembering to deal with both problems, one at a time.
It looks as though one way to get a narrower interval is to go with a lower level of confidence.
The converse is also true: to be very very sure of yourself, you will sacrifice interval size.
Fortunately the relationship between interval width and confidence level is not a linear one. This relationship is linear, meaning that an increase in variability will produce the same effect on interval size, whether variability is high or low.
Here we are again, where increasing sample size pays off big in terms of a narrower interval when sample sizes are small to begin with.
You may safely ignore the section in Chapter 5 called "Selecting an Appropriate Sample Size," on pages 139-140.
When you've finished all of the practice problems in the textbook, request a Chapter 5 worksheet via e-mail.
Understanding statistical intervals: part 2 propharma group, Part 1 of this series discussed confidence intervals.
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Shortly, I'm sure someone will post to explain to you how to do a Pythagorean Theorem calculation.
In words: The square of the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides.
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