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UNIT FOUR/CHAPTER NINE “SAMPLING DISTRIBUTIONS”

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(1) “Sampling Distribution of Sample Means” > When we take repeated samples and calculate from each one, these sample means vary. The shape, mean, and standard deviation of their graph is called the sampling distribution of sample means. This applies to having taken many, many samples – not just one or two. (2) When the population of individual values in a population is Normally distributed, then the sampling distribution of is also Normally distributed.

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(3) The sampling distribution of is less variable than that of the entire population of individual values. (For example, the yellow Post-Its on the wall form a graph that has a smaller standard deviation than the red curve of the population of individuals.) (4) The sampling distribution of sample means has two parameters with special symbols and values:

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(4) (continued) In words, this says that the mean of the sampling distribution of sample means is the same as the mean of the population of individuals. (For example, the yellow curve of Post-Its – each of these is a sample mean – has the same mean as the red curve of individuals.) In words, this says that the standard deviation of the sampling distribution of sample means is much smaller than the standard deviation of the population of individuals.

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(4) (continued) Both of these formulas are true regardless of whether the population of individuals is Normal or not. (5) Section 9-3: The Central Limit Theorem says this: > IF “n” is large (generally 30 or larger), > THEN the sampling distribution of sample means is approximately Normal, regardless of whether the population of individual values is normal, skewed, or anything else.

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Section 9-2: Similarly, suppose that we are gathering data, but the thing we are recording is not a sample mean, but a sample proportion – a p-hat. Naturally, these would vary, since p-hats are statistics. If we made a graph of these p-hats over time – taking many, many samples and recording the sample proportion each time, then these would form a shape with certain values also. (6) The mean of these p-hats would equal the population percentage – the true value of “p”. The standard deviation of these p-hats is dependent on the true value of “p” and also the value of “n” – the size of the samples from which the p-hats came.

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These p-hats – which vary, since they are statistics – can be graphed, of course, and, IF certain conditions are met, then we know what sort of shape they will make. (7) IF all of these four conditions are met: * the sample was an SRS, * the size of the population is >10n, * the value of np is >10, and * the value of n(1-p) is >10, THEN the sampling distribution of p-hat is approximately Normal. (This is similar to the Central Limit Theorem, but not called that.)

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