Sampling Theory

Domain of Deductive Thinking

From the viewpoint of epistemology, statistics subsumes two main domains. The deductive domain pertains to a description of the formal properties of the general linear model. The majority of relationships described so far were elaborated by using formal derivations and hypothetical examples made with little regard for reality. The argumentation was conducted in logical terms, using mathematical formulae to correctly move from one premise to another in a precise and sequential manner. The truth of this type of presentation has little to do with the realities outside the confines of this deductive model. A formally correct structure of the general linear model assures the transition of meaning independent of what type of problems it is used to solve. It can solve real problems, 'what-if' problems, as well as problems which are imaginary or nonsensical.

Domain of Inductive Thinking

When applied, statistics help scientists draw conclusions from observations and experiments. This is the domain of inductive reasoning. The preferred form of these conclusions is that the observed relationships are not only associations, but that the observed and described chain of events represents a causal sequence. The main objection to correlational methods is that association per se does not mean that the events are related in the causal sense.

Causality and Manipulation of the Factors

The inference of causality typically necessitates the manipulation of factors within the context of an experiment. If all factors except one are kept constant and the events systematically change as a result of manipulation of that factor, then the change can be ascribed to that particular factor (experimental condition or treatment) and some degree of causality can be inferred. 

Degrees of Confidence

The truth or falsity of any empirical observation cannot be stated with certainty that is typical of inductive arguments. As David Hume in his Enquiry Concerning Human Understanding convincingly argued, even the statement that 'the sun will not rise tomorrow is no less intelligible a proposition, and implies no more contradiction than the affirmation that it will rise. We should in vain, therefore, attempt to demonstrate its falsehood.' The question whether manifestations of observed phenomena are dependent on occurrences of measured antecedents can be answered with the aid of statistical inference with certain degrees of confidence.

The Sampling Distribution Of Means

Statistical inference assumes a population of events. Our observations or experiments were sampled from that population. What is likely to happen in the course of this sampling?

Define a Population

Let us define a population

 

 

 

shown in the figure below. The mean of this population equals 2, its variance equals .67.

 

 

All Possible Samples of a Given Size

Draw all possible samples of size 2 from the above population and compute their means.

 

 

The distribution of these sampled means can be plotted as shown below

 

   

The distribution of the sampled means approximates the binomial distribution. This fact is quite remarkable if one considers that the distribution of the population sampled from [1 2 3] is definitely rectangular. This observation also leads credence to the oft repeated statement that the sampling distribution of means is robust with respect to the assumption of normality.

1. Mean and Variance of the Sampling Distribution of Means

The mean of the population of the individual scores (Mx) is 2.

 

 

The mean of the sample means for our example can be computed as (1.0 + 1.5 + 2.0 + 1.5 + 2 + 2.5 + 2+ 2.5+3.0) / 9 which equals 2.

 

 

Thus, the mean of the sample means is the best estimate of the mean of the population, i.e.,

 

 

The variance of the sample means, , can be computed as a variance of the variable [1  1.5  2  1.5  2  2.5  2  2.5  3] and equals .33.

 

 

2. Variance of the Mean and Standard Error of the Mean

The variance of the sampling distribution of means can be also computed directly as

 

 

where n is the sample size.

The above formula is not intuitively obvious. To explain it, first, inspect the following table where the sampling process was rewritten from the matrix

 

 

into a vector form.

 

 

 

(1) Assumption of Independent

The first assumption the classic, probability based sampling theory makes is the assumption that the elements of the population sampled from are independent, that is, that they do not enter into preferential pairings. This assumption can be verified here by computing the correlation between sampled elements which is, indeed, zero.

 

Variance of the Sampled Means

As a first step to compute the variance of the sampled means, we had to sum the sampled values, as shown below.

 

 

Variance of a sum

The variance of a sum of two variables is defined as the sum of the variances of the variables being summed, plus two times their covariance.

 

   

 

However, since the summed elements are uncorrelated, there is no covariance term.

 

 

 

(2) Assumption of Homoscedascity

The second assumption made by the classical sampling theory is the assumption of homoscedascity, i.e., that the variances of the sampled elements are the same.

 

Thus the equation defining the variance of a sum of two elements, X1 and X2,  can be further simplified as 

 

 

 

Divide the sum by a constant

What happened to this variance when the sampled elements were divided by 2 to obtain the means?

 

 

One of the basic theorems of statistics is that the variance of a variable divided by a constant equals the variance of the variable divided by the square of that constant, thus

 

 

and

 

 

For the example of the sampling distribution of means, the variance of the sampled means (.33) indeed equals the variance of the population sampled from (.67) divided by the size of the sample, n, equal to 2. The square root of the above expression is also known as the standard error of the mean.

 

 

The One-Sample z Test

The one-sample z test is designed to test whether a sample statistic (e.g., mean, proportion, or correlation) is significantly different from its corresponding population parameter. 

The Critical Ratio

How far does a random sample mean differ from the mean of the sampling distribution of means in standard error units?

Form a z ratio by using the difference between a random sample mean and the mean of the population in the numerator and the standard error of the mean as its denominator.

   

The critical ratio defines a z score. Consequently,  the calculated z ratio can be associated with an unique area under the standard normal distribution and its significance can be interpreted in terms of its probability densities.

Research Question

Consider the following example. A researcher measures the body temperatures of a random sample of 64 patients diagnosed with an illness. The mean temperature of this sample turns out to be 100.6.

Population Mean and Standard Deviation

The mean temperature of the general population is 98.6 degrees Fahrenheit with a standard deviation equal to 8. (This standard deviation is fictitious, to make the computations easier.)

Can we be reasonably sure that one of the symptoms of this illness is an increase in the body temperature?

The selection of the .05 and the .01 levels as probability limens is based on consensus within the discipline. This consensus is related to empirical observations of probable and improbable outcomes, well illustrated by the following example.

Consider a simple probability event of the flip of a coin. The probability of observing one head is .50. The probability of observing two heads in two tosses is .50 x .50 = .25. The probability of observing three heads in a row is .50 x .50 x .50 = .13. The probability of observing four heads in a row is .50 x .50 x .50 x .50 = .06. The probability of observing five heads in a row is .03. The probability of observing six heads in a row is .02; of seven heads in a row .01. Between four and five heads in a row our belief in the fairness of the coin begins to waiver; more than six heads in a row makes us nearly sure that the coin is loaded. Hence, the .05 and .01 significance levels.

In sum, it is possible to obtain seven heads in a row. However, it is rare, not likely to happen.

Conduct a one-sample z test at the .05 significance level.
 

The population standard deviation is 8 and the sample size is 64. Thus, 8/8 = 1 

How far does the sample mean of 100.6 differ from the mean of the sampling distribution of means in standard error units? The z ratio is computed as

 

It means that a sample mean of 100.6 is 2 standard deviation units above the mean of the sampling distribution of means.

The probability associated with a z-ratio of 2.00 or more is .02. Since the observed probability is less than .05, the researcher would declare the result to be statistically significant. 

Report the Result

A one-sample z test was conducted to evaluate whether the mean body temperature of patients diagnosed with an illness was higher than 98.6 degrees Fahrenheit. The z test was significant, z = 2, p < .05. Patients with the diagnosed illness (M= 100.6) had higher body temperature.