Sampling and Randomization Flashcards
The Law of Large Numbers
As the sample increases, its mean is more likely to be close to the mean of the population
The Central Limit Theorem
The sampling distribution converges to a bell curve (normal distribution) as the sample gets larger
As we increase our sample size, we will increase…. (precision or accuracy)?
We will increase precision.
More accuracy comes from randomization.
Two basic questions of statistics?
How confident can you be in your results?
How big does your sample need to be? (in order to be confident in our results)
The sample distribution is likely to assume what shape? Why is this important?
The Central Limit Theorem tells us that regardless of the population distribution and the number of samples we are able to draw from the population, the sampling distribution will assume the shape of a bell curve.
This is important because with knowledge of the mean and the standard error, we can calculate the area under different portions of the bell curve. Doing so allows us to make statistical inferences about the underlying population even if we only observe one point on the sampling distribution.
If we know the population mean & standard deviation, why don’t we need to run a simulation thousands of times to determine the shape, mean and spread of our sampling distribution?
The shape of the sampling distribution need not (and usually does not) bear any resemblance to the underlying population distribution. The sampling distribution is a distribution of different sample means, whereas the “distribution” of one single sample is simply a single data point within the broader sampling distribution.
We know from the Central Limit Theorem that the sampling distribution takes the shape of a bell curve, with a mean given by the population mean and spread that is a function of the sample size and the spread of the underlying population distribution.
What’s the difference between the standard deviation and the standard error?
The standard error = the standard deviation of the sampling distribution
As our sample size goes up, the sampling distribution gets narrower…then, the standard error gets smaller but the standard deviation is unchanged
(In an individual-level randomized evaluation) Increasing the sample size by a factor of 4 results in….(what to the standard error?)
It reduces the standard error by half.
Recall that the standard error is a function of the population standard deviation and the sample size. Specifically, the standard error is given by the standard deviation divided by the square root of the sample size. Increasing the sample size from N to 4N would thus decrease the standard error by 1 divided by the square root of 4 i.e. 1/2. Quadrupling the sample size would thus halve the standard error.
Standard deviation
A measure of the spread (or dispersion) of our population
Standard error
The standard error tells us the spread of the sampling distribution which is a function of the standard deviation of the underlying population, but is also influenced by the size of our sample.
T/F: The sampling distribution may not be normal if the population distribution is skewed
False
What happens to the sampling distribution if we draw a sample size of 50 instead of 10, and plot the mean (thousands of times)? What will happen to the bell curve?
Per the Law of Large Numbers, as the sample increases, its mean is more likely to be close to the mean of the population. Thus, the sampling distribution from a larger sample (50 vs. 10) will yield a narrower bell curve, with more values closer to the actual mean.
