What should you know about Estimation and Sampling Distributions?
- You need to do a careful read through of the information presented in Chapter 7 of your text!
I present a basic overview, but your book spends more time presenting the finer details.
- As researchers, we often want to be able to generalize from our sample to the larger population. Since we are interested in what is going on the population, but generally don’t know this information, we need to estimate it from our sample information.
- A sample statistic may differ from the value of its corresponding population parameter because of sampling error.
Estimation of the Population Mean
- In the absence of any other information, the sample mean you observe is the "best estimate" of the value of the population mean
- Statisticians have determined that over all possible random samples of a given sample size from the population, the mean of the distribution of sample means will equal the population mean
- Because of this property (above), the sample mean is an unbiased estimator of the population mean
Estimation of the Population Variance and Standard Deviation
- The sample variance is a biased estimator of the population variance because it underestimates (is smaller than) the population variance across all possible samples of a given size.
- We can correct this bias by calculating the variance estimate:
- Because the standard deviation is calculated from the variance, we calculate the standard deviation estimate from the variance estimate:
Degrees of Freedom
- An important concept in statistical estimation is degrees of freedom
- Tells us the number of pieces of information "free of each other" in that they cannot be deduced from one another
- As the degrees of freedom associated with an estimate increase, the accuracy of the estimate also tends to increase
Sampling Distribution of the Mean and Central Limit Theorem
- A distribution of scores consisting of the mean for all possible samples of a given size is called a sampling distribution of the mean
- We gain insight into the mean and standard deviation of a sampling distribution of the mean, as well as its shape, from the central limit theorem
- There are 3 key implications of the central limit theorem for sampling distributions of the mean:
- The mean of a sampling distribution of the mean is always equal to the population mean (m
)
- The standard deviation of a sampling distribution of the mean is called the standard error of the mean
- The standard error of the mean reflects the accuracy with which sample means estimate a population mean. In other words, it is an index of sampling error
- The smaller the value of the standard error of the mean, the less sampling error (or, the more accurately the sample means estimate the population mean)
- The standard error of the mean is calculated from:
- We often don’t know s
, so we need to estimate the standard error of the mean:
- The shape of a sampling distribution of the mean approximates a normal distribution as the sample size on which it is based increases, regardless of the shape of the underlying population
- when the sample size is greater than 30, the normal approximation is quite good
The major points about the sampling distribution of the mean are summarized on page 197 of your textbook. You should review these!
- This information lays the foundation for the statistical tests we will be doing for the remainder of the semester
- Without this theoretical foundation, we would be limited to just describing our samples
