Estimation and Sampling Distributions

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Estimation and Sampling Distributions – Chapter 7

WARNING! This material is more abstract than previous material. However, it is extremely important for understanding what is going on the rest of the semester.

Inferential Statistics

- Research in behavioral science is usually concerned with very large populations, but is based on observations from smaller samples

Using statistics, we have ways of estimating population parameter values and determining how "close" our obtained values are to population values.

Estimation of Population Means

When we are interested in describing the population (that is the goal in inferential stats) in terms of its mean on some characteristic, we can use a sample or subset of the population to attempt to make inferences about the population mean.

- Theoretically, there is a true population mean (), but we can never find out its value

- We are trying to estimate this value by using the sample data that we have

- However, it is likely that the value that we obtain from our sample on the characteristic we are measuring will not be equal to the actual population value we are interested in.

- The fact that a sample statistic may not equal the value of its corresponding population parameter is said to be the result of sampling error.

Sampling error occurs when the sample values we obtain are different from the actual population values because they are based only on a portion of the overall population.

- The amount of sampling error can be represented by the difference between the sample statistic that we obtain (e.g., the sample mean) and the value of the population parameter that we are interested in (e.g., population mean):

- In the absence of any other information, the sample mean that one observes is the "best estimate" of the value of the population mean.

- If we took the mean of all possible sample means, this mean would equal the true population mean.

This property of the mean is what makes it an unbiased estimator of the population mean.

An unbiased estimator of a population parameter is an index whose average (mean) over all possible random samples of a given size equals the value of the actual population parameter.

- It is this property of the sample mean that makes it the best guess or estimate of the population mean.

Estimation of Population Variance and Standard Deviation

- Just as we found that our sample mean is likely to be different from the actual population mean, the same thing happens with the variance and standard deviation.

- Unlike the sample mean, however, the sample variance is not our best guess or estimate of the true population variance.

- Sample variance is a biased estimator of the population variance in that it underestimates (is smaller than) the population variance (across all possible samples of a given size).

- However, the formula for variance can be altered so as to make the sample variance an unbiased estimator of the actual population variance:

Instead of:

We would use:

Where (or "s-hat squared") is the symbol that we use to represent a sample estimate of the population variance, called the variance estimate.

The N-1 used in the denominator of the equation is a correction that makes this variance estimate an unbiased estimate of the actual population variance.

the variance estimate is thus larger than the sample variance (which corrects for underestimation)

- Estimation of the Population Standard Deviation

- The sample standard deviation is defined as the square root of the variance. So we can estimate the population standard deviation by taking the square root of

Degrees of Freedom

- An important part of inferential statistics/estimation

- The phrase degrees of freedom is used to tell the number of pieces of information that are independent, or "free of each other."

As degrees of freedom associated with an index increase, the accuracy of the estimate also tends to increase

Central Limit Theorem (stated on p. 189) is based on the sampling distribution of the mean, which is a theoretical distribution consisting of mean scores for all possible random samples of a given size that can be drawn from a population.

"The sampling distribution of the mean has a mean of , a standard deviation of , and approaches a normal distribution as the sample size on which it is based becomes larger (approaches infinity)"

Important Parts of Central Limit Theorem:

1. The mean of the sampling distribution of the mean is always equal to the population mean.

2. The standard deviation of a sampling distribution of the mean is called the standard error of the mean ()

It reflects the average deviation of the sample means from the population mean.

Standard error of the mean can be estimated from sample data, since in practice, the population standard deviation is rarely known

so we estimate the standard error of the mean by using our sample estimates of the population values:

- Formula for the standard error of the mean shows that there are two factors that influence the standard error

a) Sample size – as N gets larger, standard error gets smaller.

b) The variability of the population – as variability gets smaller, standard error decreases as well

3. The shape of the sampling distribution of the mean

- The sampling distribution of the mean approximates a normal distribution when the sample size on which the means are based is sufficiently large

- This is true regardless of the shape of the underlying population

- When we have an N or 30 or more, the approximation to the normal curve is usually pretty close.