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Introduction to Inferential Statistics

- Statistical analyses are initially divided into
- Descriptive Statistics or Inferential

Statistics. - Descriptive Statistics are used to organize

and/or summarize the parameters associated with

data collection (e.g., mean, median, mode,

variance, standard deviation) - Inferential Statistics are used to infer

information about the relationship between

multiple samples or between a sample and a

population (e.g., t-test, ANOVA, Chi Square). - The Scientific Method uses Inferential Statistics

to determine if the independent variable has

caused a significant change in the dependent

variable and if that change can be generalizable

to the larger population. - If a dependent variable within the population is

normally distributed and we can calculate or

estimate the mean and standard deviation of the

population, then we can use probabilities to

determine significance of these relationships. - To begin we must collect a representative sample

of our much larger population.

Sampling

- A Representative Sample means that all

significant subgroups of the population are

represented in the sample. - Random Sampling is used to increase the chances

of obtaining a representative sample. It assures

that everyone in the population of interest is

equally likely to be chosen as a subject. - Random Number Generators or Tables are used to

select random samples. Each number must have the

same probability of occurring as any other

number. The larger the random sample, the more

likely it will be representative of the

population.

Sampling Distribution

- Generalization refers to the degree to which the

mean of a representative sample is similar to or

deviates from the mean of the larger population.

This generalization is based on a distribution of

sample means for all possible random samples. - This distribution is called a Sampling

Distribution. - It describes the variability of sample means that

could occur just by chance and thereby serves as

a frame of reference for generalizing from a

single sample mean to a population mean. - It allows us to determine whether, given the

variability among all possible sample means, the

one observed sample mean can be viewed as a

common outcome (not statistically significant) or

as a rare outcome (statistically significant). - All possible random samples of even a modest size

population would consist of a very large number

of possibilities and would be virtually

impossible to calculate by hand. Therefore we use

statistical theory to estimate the parameters.

Sampling Distribution

- The Central Limit Theorem states that the

distribution of sample means approaches a normal

distribution when n is large. - In such a distribution of unlimited number of

sample means, the mean of the sample means will

equal the population mean.

Standard Error of the Mean

- The standard deviation of the distribution of

sample means is called the Standard Error of the

Mean or Standard Error for short. It is

represented by the following formula - Since the standard deviation of the population is

often unavailable, a good estimate of the

standard error uses the standard deviation of the

sample (s). This newly modified formula of the

Standard Error is shown below

Statistical Significance

- Now that we have the parameters of the Sampling

Distribution we can see how to use this

distribution to determine if the mean difference

between two samples or between a sample and a

population are significantly different from each

other. - The Research Hypothesis (H1) states that the

sample means of the groups are significantly

different from one another. - The Null Hypothesis (H0) states that there is no

real difference between the sample means. - Anytime you observe a difference in behavior

between groups, it may exist for two reasons - 1.) there is a real difference between the

groups, or - 2.) there is no real difference between the

groups, the results are due to error involved in

sampling. - This error can be described in two ways
- Type I error is when you reject the null

hypothesis when shouldn't have - Type II error is when you fail to reject the null

hypothesis when you should have

Statistical Significance

- The probability of committing a Type I error is

designated by alpha. - An alpha level of 0.05 is reasonable and widely

accepted by all scientists. - The null hypothesis can be rejected if there is

less than 0.05 probability of committing a Type I

error ( p lt .05 ).

Two-Tailed Hypothesis

- If the research hypothesis does not predict a

direction of the results, we say it is a

Two-Tailed Hypothesis because it is predicting

that alpha will be split between both tails of

the distribution. If the sample mean falls in

either of these areas we can reject the null

hypothesis. This is shown below

One-Tailed Hypothesis

- If the scientific hypothesis predicts a direction

of the results, we say it is a One-Tailed

Hypothesis because it is predicting that alpha

will fall only in one specific directional tail.

If the sample mean falls in this area we can

reject the null hypothesis. This is shown below