Last chapter we talked about the probability of finding a particular score, or set of scores in the population. Now, we will instead talk about the probability of finding particular samples in the population.
This is getting closer to inferential statistics. Recall that the goal of inferential statistics is to make claims about population parameters based on sample statistics.
So the logic will be something like this. We can't measure the whole population, so we take a sample. Our best estimate for the mean of the population will be the mean of our sample. (remember that it is only an estimate because we have sampling error before - the difference between a sample statistic and the corresponding population parameter)
It sounds simple and straight forward, but consider the following:
Suppose that you take 3 different samples from the same population. They are going to be different from one another. They will have different shapes, different means, and different variability. So how do you figure out what the best estimate of the population mean is?
How many possible samples can we take? Infinite (remember that we are sampling with replacement) Luckily for us, the huge set of possible samples forms a simple, orderly, and predictable pattern (a sampling distribution). Because of this, we are able to base our predictions about sample characteristics on the distribution of sample means.
A sampling distribution is a distribution of statistics obtained by selecting all the possible samples of a specific size from a population.
In other words, what we want to do is look at all of the possible samples (of a particular size, this part is important) and make predictions based on the properties of all of them.
Let's look at a more concrete example (the book's example):
because this population is so small we actually can know the mean (and variability): m = 5, but suppose that we didn't, and wanted to be able to make an estimate based on sampling
step 1: pick a sample size: for this example we'll pick samples of n = 2 - we'll talk more about sample size a little later, but typically the bigger your sample size, the more likely that your samples will be similar to one another (and to the population as a whole)
step 2: now consider all of the possible samples that you could get, and look at their distribution
____________________________________ scores sample mean sample first second () 1 2 2 2 2 2 4 3 3 2 6 4 4 2 8 5 5 4 2 3 6 4 4 4 7 4 6 5 8 4 8 6 9 6 2 4 10 6 4 5 11 6 6 6 12 6 8 7 13 8 2 5 14 8 4 6 15 8 6 7 16 8 8 8
Distribution of the sample means
f 2 1 3 2 4 3 5 4 6 3 7 2 8 1 |
step 3: Now you're ready to answer questions like: What is the probability of getting a sample with a mean greater than 7? p( > 7) = ?
In our example we've simplified things greatly. We have a really small population, and we took a pretty small sample. Most situations, however, will be much more complex. Lucky for us there are some properties of means of samples that will help us out.
mean:
Look at our example, the expected value of (the mean of the sample means) is:
2 + 3 + 4 + 5 + 3 + 4 + 5 + 6 + 4 + 5 + 6 + 7 + 5 + 6 + 7 + 8 = 80 = 5.0 16 16
standard error of = = standard distance between and m .
the major purpose/use of the standard error of is that it tells us how well the sample mean estimates the population mean. In other words, how big is the sample error.
the numerical value of the standard error is determined by two characteristics: the variability of the population & the size of the sample
large s big differences from the pop mean |
small s small differerences from the pop mean |
2) the size of the sample - the larger your sample size (n), the more accurately the sample represents the population. This is known as the law of large numbers .
- If I randomly selected 1 student, how accurately will that student's score predict the population's score? | |
- Suppose that I take 5 students. Are things more accurate? | |
- what about 100 students? |
these two characteristics are combined in the formula for standard error.
standard error of = =
All of these properties (shape, mean, variability) are covered in the Central Limit Theorem
So, when n is large (at or above 30): ~ N (m , )
Now let's bring probability back into the picture.
Example:
First we need to get the distribution of the samples (note: we'll assume a normal distribution even though n is less than 30.) ~ N (m, ) = N(100, 5)
Now we need to figure out the z-score that corresponds to this sample mean:
the z-score pretty much looks like what we've used before: Z =
P( > 112) = P(Z > (112 - 100)/ 5 ) = P(Z > 2.4) = 0.0082
- at first it looks wrong - it seems like 112 should be less than a z = 1, because 115 is where z should equal 1 |
|
- however, we must remember that this isn't the correct distribution to be looking at, we need to look at the distribution of sample means. -we know that the distribution of sample means has a standard error = 5 and a mean = 100. - So 112 should have a z >2 |
Example:
Now we need to figure out the mean that corresponds to this range:
step 1: look at unit normal table for 90%
step 2: = 1.28 * + 100 = (1.28)(3) + 100 = 103.84
Suppose that we asked the same question for a smaller sample, n = 16? How does the answer change?
step 1: look at unit normal table for 90%
step 2: = 1.28 * + 100 = (1.28)(3.75) + 100 = 104.80
so, for a group of 16 people, they'd have to have a mean of over 104 to be in the top 10%
What about other sample sizes?
n = 9
So the take home message is: the smaller your sample size, the larger your sampling error (standard error, ).
Interpreting sampling variance:
A) Sampling Error: any one sample may over or under estimate (with about half of each).
B) Standard Error: most of the means will be close to m , but some are further away. The variability of these sample means represents the standard distance between m and, or the "standard" error distance. It defines the relationship between sample size and the accuracy with which represents m .
We've sort of talked about these two, but not by name:
C) Reliability: As the standard error gets smaller, then our confidence in as
an estimate of m increases.
- We've seen this in our examples. As n gets larger, the sample statistics become better estimates of the population parameters. So, repeated samples of a large n, will usually have similar statistics (all near the population parameters).
D) Stability: The smaller the standard error is, the less likely adding or deleting
or changing a single score would chage the estimate of m .
consider the population X ~ N(50, 10) [m = 50; [s = 10]
suppose that we take two samples:
So sample 2 is morestable to change than sample 1. More generally, the smaller the standard error, the more stable the sample.