In large populations, it is not practical to take a census, so instead, sampling is used to infer something about the larger population. In order to do that, we need to know how close the sample mean x̅ is likely to be to the population mean μ. As we know, different random samples will yield different values of x̅. The question then becomes, how can we describe this sampling distribution of possible x̅-values? We can think of x̅ as a random variable because it takes numerical values that describe the outcomes of the random sampling process. Because of that, we can use what we learned in the last unit to examine the probability distribution of x̅.
A parameter is a number that describes some characteristic of the population.
A statistic is a number that describes some characteristic of the sample.
μ represents population mean (the parameter), and x̅ represents represents sample mean (the statistic).
p represents population proportion (the parameter), and p̂ represents sample proportion (the statistic).
The sampling variability describes how a value of a statistic varies in repeated random sampling.
The sampling distribution is the distribution of values taken by the statistics in all possible samples of the same size from the same population. The sampling distribution displays a collection of the values of p̂ from all of the possible samples of the same size. The population distribution gives the values of the variable for all individuals in the population.
The distribution of sample data shows the values of the variable for the individuals in the sample.
A statistic used to estimate a parameter is an unbiased estimator if the mean of its sampling distribution is equal to the value of the parameter being estimated.
Important note: “Unbiased” doesn't mean perfect. An unbiased estimator will almost always provide an estimate that is not equal to the value of the population parameter. It is called “unbiased” because the estimates won't consistently be too high or too low. In other words, the center of the statistic’s sampling distribution will be in the right place.
The sample proportion p̂ from a random sample of any size is an unbiased estimator of the parameter p. As we can see from the previous example, though, larger random samples have a clear advantage. They are much more likely to produce an estimate close to the true value of the parameter. In other words, larger random samples give us more precise estimates than smaller random samples. Low variability is always best.
Taking a larger sample does not fix bias however. Remember that even a very large voluntary response sample or convenience sample is worthless because of bias.
The variability of a statistic is described by the spread of its sampling distribution. This spread is determined mainly by the size of the random sample. Larger samples give smaller spreads. The spread of the sampling distribution does not depend much on the size of the population, as long as the population is at least 10 times larger than the sample.
Bias (center), variability (spread), and shape: We can think of the true value of the population parameter as the bullseye on a target and of the sample statistic as an arrow fired at the target. Both bias and variability describe what happens when we take many shots at the target.
Bias means that our aim is off and we consistently miss the bullseye in the same direction. High variability means that repeated shots are widely scattered on the target. Ideally, we’d like our estimates to be accurate (unbiased) and precise (have low variability).
Summary
- Shape: In some cases, the sampling distribution of p̂ can be approximated by a Normal curve. This depends on both the sample size of n and the population proportion p.
- Center: The mean of the distribution is μp̂ = p This makes sense because the sample proportion p̂ is an unbiased estimator of p.
- Spread: For a specific value of p, the standard deviation σp̂ gets smaller as n gets larger. The value of σp̂ depends on both n and p. As sample size increases, spread decreases.
The mean of the sampling distribution of p̂ is μp̂ = p The standard deviation of the sampling distribution of p̂ is σp̂ = √(p(1-p))/n) as long as the 10% condition (n ≤ 1/10N) is satisfied.
As n increases, the sampling distribution of p̂ becomes approximately Normal. Before you perform Normal calculations, check that the Large Counts condition is satisfied: np ≥ 10 and n(1-p)≥10