## Chapter 7 - Probability and Samples: The Distribution of Sample MeansThe topics discussed in this chapter are some of the most important topics for your understanding of inferential statistics. They also happen to be the most conceptually difficult to understand. Here is where we begin to discuss the theoretical tools that we use to test the significance of our research. As usual, begin with the Preview and Overview. We are working with samples from the population now, and one of the major issues is how to determine whether our sample is truly representative of the population. The difference between the mean of our sample and the population mean is called sampling error. We want to reduce sampling error as much as possible, and the the law of large numbers tells us that we can do this by having a large sample. One of the main goals of statistics is to determine the probability that our sample is representative of the population and a large sample is more likely to be representative of the population than is a small sample. Section 7.2 begins the discussion of sample means
(the mean of a sample). It should be clear that if you repeatedly take
a sample from a population and calculate the means of those samples, the
means will likely not all be the same (unless your population has no variability).
If you take all the sample means and plot them on a histogram, you will
have created what is called a distribution of sample
means, sometimes just referred to as a sampling
distribution. In other words, if you repeatedly took samples of
size Look closely at Example 7.1. We will do this in class as well. The distribution of sample means has some interesting characteristics.
First, if your samples are big enough (a large Read and understand everything in Section 7.2 (try the Learning Check) before moving on to Section 7.3. Section 7.3 reintroduces probability. If you have a distribution of sample
means, and you know that it is approximately normally distributed, you
can find the probability of obtaining any particular sample mean using
the same techniques that we used in the last chapter for an individual
score from a population of scores. First you have to convert the sample
mean into a Again, the Learning Check can be very helpful. These previous two sections, 7.2 and 7.3, are If we did this, the shape of the sampling distribution of the means would allow us to make some statements about the probability of getting any particular sample mean. Obviously, sample means that are far from the population mean would be unlikely, whereas sample means that are close to the population mean would be more probable. It is interesting to note that the distribution of sample means is shaped
like a normal distribution (provided your sample Now, if we want to know the probability of getting a particular sample
mean, given that we know the population mean, all we have to do is find
out how many standard deviations (now called "standard errors")
our sample mean is away from the population mean. In other words, we have
to compute the I can not stress enough that this is one of the most important chapters of the entire text. ## ProblemsMany of these will be useful tests of your understanding. Once again, I think that you should be able to do every problem in this chapter, but if your time is limited, try the odds. |