Chapter 5 Introduction to Inferential Statistics

Definition zinfer - vt., arrive at a decision by or opinion by reasoning from known facts or evidence.

Sample A sample comprises a part of the population selected for a study.

Random Samples If every score in the population has an equal chance of being selected each time you chose a score, then it is called a random sample. Random samples, and only random samples, are representative of the population from which they are drawn.

EVERY Q: ON WHAT MEASURES IS A RANDOM SAMPLE REPRESENTATIVE OF THE POPULATION? A: ON EVERY MEASURE.

REPRESENTATIVE ON EVERY MEASURE zThe mean of the random sample’s height will be similar to the mean of the population. zThe same holds for weight, IQ, ability to remember faces or numbers, the size of their livers, self-confidence, how many children their aunts had, etc., etc., etc. ON EVERY MEASURE THAT EVER WAS OR CAN BE.

All sample statistics are representative of their population parameters zThe sample mean is a least squares, unbiased, consistent estimate of the population mean. zMS W is a least squares, unbiased, consistent estimate of the population variance.

REPRESENTATIVE ON measures of central tendency (the mean), on measures of variability (e.g., sigma 2 ), and on all derivative measures zFor example, the way scores fall around the mean of a random sample (as indexed by MS W ) will be similar to the way scores fall around the mean of the population (as indexed by sigma 2 ).

THERE ARE OCCASIONAL RANDOM SAMPLES THAT ARE POOR REPRESENTATIVES OF THEIR POPULATION zBut 1.) we will take that into account zAnd 2.) most are fairly to very good representatives of their populations

Population Parameters and Sample Statistics: Nomenclature The characteristics of a population are called population parameters. They are usually represented by Greek letters ( ,  ). The characteristics of a sample are called SAMPLE STATISTICS. They are usually represented by the English alphabet (X, s).

Three things we can do with random samples zEstimate population parameters. This is called estimation research. zEstimate the relationship between variables in the population from their relationship in a random sample. This is called correlational research. zCompare the responses of random samples drawn from the same population to different conditions. This is called experimental research.

Estimating population parameters zSample statistics are least squares, unbiased, consistent estimates of their population parameters. zWe’ll get to this in a minute, in detail.

Correlational Research We observe the relationship among variables in a random sample. We are unlikely to find strong relationships purely by chance. When you study a sample and the relationship between two variables is strong enough, you can infer that a similar relationship between the variables will be found in the population as a whole. This is called correlational research. For example, height and weight are co-related.

Another way to describe correlational research zA key datum in psychology is that individuals differ from each other in fairly stable ways. zFor example, some people learn foreign languages easily, while others find it more difficult. zCorrelational research allows us to determine whether individual differences on one variable (called the X variable) are related to individual differences on a second variable (called Y).

What is to come- CH. 6 & 7 zIn Chapter 6, you will learn to turn scores on different measures from a sample into t scores, scores that can be directly compared to each other. (You will also learn to use the t distribution to create confidence intervals and test hypotheses.) zIn Chapter 7, you will learn to compute a single number that describes the direction and consistency of the relationship between two variables. That number is called the correlation coefficient.

What is to come – CH. 8 zIn Chapter 8, you will learn to predict scores on one variable scores on another variable when you know (or can estimate) the correlation coefficient. zIn Chapter 8, you will also learn when not to do that and to go back to predicting that everyone will score at the mean of their distribution.

What is to come - CH 9 – 11 Experimental Research In Chapters 9 – 11 you will learn about experiments. In an experiment, we start with samples that can be assumed to be similar and then treat them differently. Then we measure response differences among the samples and make inferences about whether or not similar differences would occur in response to similar treatment in the whole population. For example, we might expose randomly selected groups of depressed patients to different doses of a new drug to see which dose produces the best result. If we got clear differences, we might suggest that all patients be treated with that dose.

In this chapter, we will focus on estimating population parameters from sample statistics.

Estimation research We measure the characteristics of a random sample and then we infer that they are similar to the characteristics of the population. Characteristics are things like the mean and standard deviation. Estimation underlies both correlational and Experimental research.

Definition A least square estimate is a number that is the minimum average squared distance from the number it estimates. We will study sample statistics that are least squares estimates of their population parameters.

Definition An unbiased estimate is one around which deviations sum to zero. We will study sample statistics that are unbiased estimates of their population parameters.

Definition A consistent estimator is one where the larger the number of randomly selected scores underlying the sample statistic, the closer the statistic will tend to come to the population parameter. We will study sample statistics that are consistent estimates of their population parameters.

The sample mean The sample mean is called X-bar and is represented by X. X is the best estimate of , because it is a least squares, unbiased, consistent estimate. X =  X / n

Estimated variance The estimate of  2 is called the mean squared error and is represented by MS W. It is also a least squares, unbiased, consistent estimate. SS W =  ( X - X) 2 MS W =  ( X - X) 2 / (n-k)

Estimated standard deviation The estimate of  is called s. s = MS W

In English zWe estimate the population mean by finding the mean of the sample. zWe estimate the population variance (sigma 2) with MS W by first finding the sum of the squared differences between our best estimate of mu (the sample mean) and each score. Then, we divide the sum of squares by n-k where n is the number of scores and k is the number of groups in our sample. zWe estimate sigma by taking a square root of MS W, our best estimate of sigma 2.

Estimating mu and sigma – single sample S# A B C X684X684 MS W = SS W /(n-k) = 8.00/2 = 4.00 s = MS W = 2.00 (X - X) (X - X) X 6.00  X=18 N= 3 X=6.00  ( X-X)=0.00  ( X-X) 2 =8.00 = SS W

Group X MS W = SS W /(n-k) = s = MS W = (X - X) (X - X)  (X-X 1 )=0.00  (X-X 1 ) 2 = Group  (X-X 2 ) 2 =  (X-X 2 )= Group X X 1 = X 2 =  (X-X 3 ) 2 =  (X-X 3 )= X 3 = /9 = = 12.32

Why n-k? zThis has to do with “degrees of freedom.” zAs you saw last chapter, each time you add a score to a sample, you pull the sample statistic toward the population parameter.

Any score that isn’t free to vary does not tend to pull the sample statistic toward the population parameter. zOne deviation in each group is constrained by the rule that deviations around the mean must sum to zero. So one deviation in each group is not free to vary. zDeviation scores underlie our computation of SS W, which in turn underlies our computation of MS W.

n-k is the number of degrees of freedom for MS W zYou use the deviation scores as the basis of estimating sigma 2 with MS W. zScores that are free to vary are called degrees of freedom. z Since one deviation score in each group is not free to vary, you lose one degree of freedom for each group - with k groups you lose k*1=k degrees of freedom. zThere are n deviation scores in total. k are not free to vary. That leaves n-k that are free to vary, n-k degrees of freedom MS W, for your estimate of sigma 2. zThe precision or “goodness” of an estimate is based on degrees of freedom. The more df, the closer the estimate tends to get to its population parameter.

Group X MS W = SS W /(n-k) = s = MS W = (X - X) (X - X)  (X-X 1 )=0.00  (X-X 1 ) 2 = Group  (X-X 2 ) 2 =  (X-X 2 )= Group X X 1 = X 2 =  (X-X 3 ) 2 =  (X-X 3 )= X 3 = /9 = = 12.32

More scores that are free to vary = better estimates: the mean as an example. Each time you add a randomly selected score to your sample, it is most likely to pull the sample mean closer to mu, the population mean. Any particular score may pull it further from mu. But, on the average, as you add more and more scores, the odds are that you will be getting closer to mu..

Book example Population is 1320 students taking a test.  is 72.00,  = 12 Unlike estimating the variance (where df=n-k) when estimating the mean, all the scores are free to vary. So each score in the sample will tend to make the sample mean a better estimate of mu. Let’s randomly sample one student at a time and see what happens.

Test Scores FrequencyFrequency score Sample scores: Standard deviations Scores Mean 87 Means:

Consistent estimators This tendency to pull the sample mean back to the population mean is called “regression to the mean”. We call estimates that improve when you add scores to the sample consistent estimators. Recall that the statistics that we will learn are: consistent, least squares, and unbiased.

A philosophical point zMany intro psych students wonder about psychologists efforts to understand and predict how the average person will respond to specific situations or conditions. zThey feel that people are too different for us to really determine such laws.

While psychology as a “science” can be criticized on a number of grounds, this is not one of them. zThe random differences among individuals form one of the bases of our science.

The effect of individual differences zAs you know, each time you add a score that is free to vary to a sample, the sample statistics become better estimates of their population parameters. zThis happens, because individuals differ randomly. So, each person’s score corrects the sample statistics back towards their population parameters. zThese include the mean, standard deviation and other statistics that you have yet to learn, such as r, the correlation coefficient that describes the strength and direction of a relationship between two variables.

Individual differences are central in correlational research zIn correlational research, we compare individuals who differ on two variables. zWe see whether differences on one variable are related to differences on the other. zIf individuals did not have (reasonably) stable differences from each other, there could be no such correlational research.

Individual differences underlie experimental research zIn experimental work, we need to do two things. zFirst, we need to compose groups that are similar to each other. zThe key to this is randomly selecting members for each experimental group. zThat way, as you add individuals who randomly differ to each group, the groups increasingly resemble the population (and, therefore, each other) in all possible regards.

Second, in experimental work, we examine the differences among the means of experimental groups. Before extrapolating to the population from differences observed among group means, we must be sure that we are not simply seeing the results of sampling fluctuation.

We must have an index of how much variation among the means simply reflects sampling fluctuation. In most of the designs we will study in Chapters 9-11, MS W tells us how much variation among the means we should have simply from sampling fluctuation. Individual differences play a large part in determining MS W.

Thus, rather than making a science of human behavior impossible, the fact that individuals differ plays a critical role in the research designs and statistical tools that have been developed.