Statistics Essentials For Dummies (17 page)
= 2. Now find P(
Z
> 2).
4. Proceed as you usually would for any normal distribution. That is, do Steps 4 and 5 described in the earlier section "Finding Probabilities for X."
For the coin flips, P(
X
> 60) = P(
Z
> 2.00) = 1 - 0.9772 = 0.0228. The chance of getting more than 60 heads in 100 flips of a coin is about 2.28 percent.
When you use the normal approximation to find a binomial probability, your answer is an approximation (not exact), so be sure you state that. Also show that you checked the necessary conditions for using the normal approximation.
Chapter 6
:
Sampling Distributions and the Central Limit Theorem
In This Chapter
Understanding the concept of a sampling distribution
When you take a sample of data, it's important to realize the results will vary from sample to sample. Statistical results based on samples should include a measure of how much they expect those results to vary from sample to sample. This chapter shows you how to do that by couching everything in terms of the sample means (for numerical data) and applying the same ideas to sample proportions (for categorical data).
Sampling Distributions
Suppose everyone on the planet rolled a single die and recorded the outcome,
X
. With all those outcomes, we'd have an entire population of values. The graph of these outcomes in the population would represent the distribution of
X
. Now suppose everyone rolled their die 10 times (a sample of size 10) and recorded the average,. With all those averages, we'd get an entirely new population — the population of sample means. The graph of this new population would represent the sampling distribution of
.
When you're talking about a particular sample mean, use the notation
. When you're talking about the random variable representing any sample mean in general, use the notation
.
A
distribution
is a listing or graph of all possible values of a random variable or a population (such as
X
) and how often they occur. For example, if you roll a fair die and record the outcome and repeat an infinite number of times, the distribution of
X
= the outcome, with numbers 1, …, 6 appearing with equal frequency. The distribution of
X
in this case is shown in Figure 6-1a.
Now apply this idea to sample means. Take a sample of values from your random variable
X
(your population), find the mean of the sample, and repeat over and over again. You now have a new random variable called, which takes on a wide range of possible values and has its own distribution.
A listing or graph of all possible values of the sample mean and how often they occur is called the
sampling distribution of the sample mean
. For example if you roll a die 10 times, find the average, and then repeat infinite times, the average will take on values fairly close to 3.5 (halfway between 1 and 6) with values near 3.5 occurring more often than values near 1 or 6. Figure 6-1b shows the actual sampling distribution of, the average of 10 rolls of a die.
is used because data represent averages based on samples, not individual values from a pop-ulation. As with any other distribution, a sampling distribution has its own shape, center, and measure of variability — the following sections discuss these features.