Inferential Psychology:
Hypothesis testing and the search for truth

Asking questions of nature has been a part of science from the beginning. In psychology we generally move towards a model of natural science that makes use of inferential hypothesis testing as a central tenet or dogma.

As indicated earlier, with respect to sampling from populations, we use a collection of probability distributions and inferential statistics to help us in making decisions about our hypotheses.

Essentially statistics are a tool that we use in making decisions.

Sometimes we use these tools thoughtfully, considering their strengths and weaknesses, sometimes we use them blindly allowing them to dominate our thoughts and remove our interpretations from the role they must play.

Remember that there are numerous assumptions behind the use of statistics in guiding our decision-making, they do not make science and truth for us.

The context of these statistical methods comes from the re-birth of positivism in the 1930s-1940s

See Cozby Chapter 13

Inferential Statistics arises from the sampling theory that makes inferences about populations based upon samples taken from such populations. They use probability distributions to estimate the likelihood that the sample results represent the population.

Random sampling is a technique that is used in attempt to create equivalent groups where there is an even playing field for the testing of hypotheses (the removal of extraneous factors-Random Error).

When performing inferential statistics we bet against Random Error that our hypothesis is correct.

Null vs. Research (alternate) Hypothesis
Null hypothesis is a statement that indicates there is no difference between the sample and the population (or the population means for the treatment and experimental groups).

Research Hypothesis Suggests that the population means are actually different either in a specific direction (one tailed) or in either direction (two tailed).

Statistical Significance is an arbitrary level of acceptance w.r.t. the likelihood that the observed differences are due to random (or measurement) error.

Decision making Type I vs. Type II Errors

When making decisions we may wish to err on the side of caution (like sentencing a possibly innocent person to death or diagnosis of fatal disease).

The use of probabilities helps us in making decisions when we know the chances of hits, false alarms, correct rejections or misses.

        True State of Affairs (what is)

Graphic version

a is the significance level of the test which indicates the probability of rejecting Ho when in fact it is true (assume difference when there is none). This is the probability of a type I error or a false alarm. 

Probability and Sampling distributions

As indicated earlier, sampling distributions have been used to estimate probabilities that samples represent their populations, observed frequencies are different from expected ones or that group means are different (and likely due to experimental treatments or manipulations).
 
  Families of sampling distributions exist,
binomial, t, normal, X², F 
 

Sample Size has impact where larger samples give better estimates of the populations thus a greater likelihood of finding significance difference if there is one.

T test is used where the observed data is considered against the likelihood that the results occurred by chance (random error).

The statistic divides group differences (difference between two group means) by the variability (Standard Deviation) within two groups combined.

T statistic = Mean1 - Mean2 divided by the
Square Root of ( Variance1/N1 + Variance2/N2)

Degrees of freedom is a number that represents the total number of scores that must be know (along with the mean) to predict with certainty one score. (the number of scores that are free to vary once the mean is known - to predict last score). Df= n1 + n2 - 2

One tailed vs. Two tailed tests
1-tail is when you specify the direction of the relationship (p will be bigger than q) and is more sensitive (requires a smaller t for same level of significance.

2-tailed is when you do not predict direction, only that there will be a difference. Here the alpha is spread out on both tails of the distribution, requiring a larger t for significance.
  Compare

Normal distribution (Z) is related to the t distribution where t approximates Z as sample size increases. Also: t²   = Z² / Z² = F

F-Test
Is related to t distribution and is used for multiple groups. While is good for only two groups ANOVA (F-test) can be done with multiple independent and/or dependent variables.

In actuality F = t² and F = (X²_v1/v1)/ X²_v2/v2) = S²1/S²2
which is the Systematic variance / Error variance

Systematic Variance is the deviation of group means from the grand mean (overall mean-of all individual scores) this is also known as between groups variance.

Error Variance is the variation of individual scores in each of the groups, also known as the within groups variance.

Calculating the Effect Size ( for a t-test) can be done by finding the square root of (t² / t² + df)

Statistical Significance helps in the task of decision making arbitrarily set at 95% confidence (p= .05) giving interval up to level of t score.

Larger samples give better estimate of populations and usually give significance for large effect size (when within group variance is low).