9.1: Sampling Distributions
One examines samples in order to come to reasonable conclusions about the population from which the sample is chosen. One must be statistically literate in order to glean meaningful information from a sample. This involves an awareness of what the sample results tell us, along with what they don't tell us. A statistic calculated from a sample may suffer from bias or high variability, and hence not represent a good estimate of a population parameter.
Parameter: An index that is related to a population.
Statistic: An index that is related to a sample.
Sampling distribution of a statistic: The distribution of values of a statistic taken from all possible samples of a specific size.
A statistic is unbiased if the mean of the sampling distribution is equal to the true value of the parameter being estimated.
Note: A sample standard deviation is not an unbiased estimator of the population standard deviation.
9.2: Sample Proportions
The normal distribution curve is often extremely useful in analyzing sample proportions. This section provides insights into the circumstances that allow for use of normal distribution properties.
If we choose an SRS of size n from a large population with population proportion p having some characteristic of interest, and if p(hat) is the proportion of the sample having that characteristic, then
•The sampling distribution of p(hat) is approximately normal.
•The mean of the sampling distribution is p (the population parameter).
•The standard deviation of the sampling distribution is sqrt[p(1-p)/n].
It is reasonable to use the above statements when
•the population is at least 10 times as large as the sample.
•np is at least 10 and n(1-p) is at least 10.
9.3: Sample Means
This section contains one of the most important of all statistical theorems, the Central Limit Theorem of Statistics. It also emphasizes that it is conventional the Greek letters µ and sigma are used for the population parameters mean and standard deviation, and that x(bar) and s conventionally represent the mean and standard deviation for samples.
