Statistics Chapter 8 Psyc 295

Comparison of 1-tailed test vs. 2 tailed tests

-1 tailed test allows you to reject the null hypothesis when the difference is relatively small, provided the difference is in the specified direction +/-. -2 tailed test requires a relatively large difference independent of direction; the sign doesn't matter!

Number of scores in the sample

-Influences the size of the standard error. -& appears in the denominator of the z-scores.

Alternative or Research Hypothesis is

1 tailed, or directional.

There are 2 possible outcomes:

1. If sample statistic, z, is located in the critical region, the null hypothesis is rejected. 2. If the sample statistic, z, is not located in the critical region, the researcher fails to reject the null hypothesis, which means they accept the null hypothesis as true.

Null Hypothesis is

2 tailed, which is non-directional.

1. ~State hypothesis. 2. predict the expected characteristics of the sample based on the hypothesis & set the criteria for a decision.

3. Obtain a random sample from the population, collect data & compute sample statistics. 4. Compare the sample data with the prediction & decide if it's: a: consistent, then the hypothesis is reasonable. or b. if it's discrepant, the hypothesis is rejected.

Alpha .01 is a 1% error or chance influenced your outcome.

95% in middle & 5% in the tail establishes scores falling within 2 standard deviations from the mean. -on the negative side it shows the decrease of scores. -& on the positive side it shows an increase of scores.

How is the value of z influenced by : An increase in the difference between the sample mean & the original population mean?

A larger difference between the mean (M) & mew (u) means a larger numerator which = a larger z score.

How is the value of z influenced by : An increase in the number of scores in the sample?

A larger sample means a smaller standard error of means which means the z-score would be larger. ???

How is the value of z influenced by : An increase in the population standard deviation?

A larger standard deviation produces a larger standard error of mean which means the z-score area that your number could fall within would be smaller. ~The denominator would be bigger.?? Kimmie?? ~This reduces the likelyhood of rejecting the null, or, in other words you accept that the hypothesis that the treatment has no effect.???

Step 1, establish hypothesis, Step 2. establish criteria for decision which is the boundary of critical region

Alpha level .05 is how much error or chance influenced my outcome as opposed to the treatment.

There's always an outside chance the jury's wrong though! Need Room for error.

Always start with the hypothesis that the treatment won't have an effect. -Keep in an unbiased state of mind; Ho = null. -H1 means treatment will have an effect.

Data is *always* collected *after* hypothesis is stated.

Data is *always* collected *after* criteria for decision is set. --This sequence assures objectivity!

In a 1 Tailed Z-crit the critical value is directional, so it is going to be *either* positive or negative to show if there's an increase or decrease.

Find the value in the tail column: 5%-->.05 = 1.65 (+ or -) 1%-->.0100 = 2.33 (+ or -) .01%-->.0010 = 2.10 (+ or -)

Alpha level

For a hypothesis test is the probability that the test will lead to a Type I error. That is, the alpha level determines the probability of obtaining sample data in the critical region even though the null hypothesis is true. The larger the alpha level for the test, e.g. 05. vs .01, the greater the power.

Significant, or statistically significant

If it is very unlikely to occur when the null hypothesis is true. That is, the result is sufficient to reject the null hypothesis. Thus, a treatment has a significant effect if the decision from the hypothesis test is to reject H0.

Null hypothesis is innocent until proven guilty! -gather evidence. --if there's sufficient evidence then the innocent claim is rejected!

If statistics exceed the table value then we can conclude our treatment had an effect & reject the hypothesis.

Alternative Hypothesis, H1, states that there is a change, a difference, or a relationship for the general population.

In the context of an experiment, H1 predicts that the independent variable, or the treatment, will have an effect on the dependent variable.

Variability of the scores

Influences size of the standard error & -Appears in the denominator of the z-score.

Alpha level, or the level of significance

Is a probability value that is used to define the concept of "very unlikely" in a hypothesis. e.g. .5%, 1% or .1%

Hypothesis test

Is a statistical method that uses sample data to evaluate a hypothesis about a population.

Critical region

Is composed of the extreme sample values that are very unlikely, as defined by the alpha level, to be obtained if the null hypothesis is true. The boundaries for the critical region are determined by the alpha level. If sample data fall in the critical region, null hypothesis is rejected.

Effect size

Is intended to provide a measurement of the absolute magnitude of a treatment effect, independent of the size of the sample (s) being used. The larger the effect size, the greater the power.

Larger z-score value = a more conservative choice.

It bumps the boundary a little further out there..

Type II error

Occurs when a researcher fails to reject a null hypothesis that is really false. In a typical research situation, means that the hypothesis test has failed to detect a real treatment effect.

Type I error

Occurs when a researcher rejects a null hypothesis that is actually true. In a typical research situation, means that the researcher concludes that a treatment does have an effect when, in fact, it has no effect.

Power

Of a statistical test is the probability that the test will correctly reject a false null hypothesis. That is, power is the probability that the test will identify a treatment effect if one really exists.

In a 2 Tailed Z-crit the critical value is divided equally in each tail of the distribution.

So, if the Alpha is 5% you would move the decimal two points, which is 0.050, divide it in 2 which is .025, then find the z-score for .2500 which is 1.96. ~%5 or .05 = + or - 1.96 ~1% or .01 = + or - 2.58 ~.1% or .001 = + or - 3.30

Null hypothesis (H0)

States that in general population there is no change, no difference, or no relationship. In the context of an experiment, H0 predicts that the independent variable, aka the treatment, has no effect on the dependent variable, which are the scores, for the population.

Alternative hypothesis H1

States that there is a change, a difference, or a relationship for the general population. In the context of an experiment, H1 predicts that the independent variable, the treatment, does have an effect on the dependent variable.

The Alpha level for a hypothesis test is the probability that the test will lead to a Type 1 error.

That is, the alpha level determines the probability of obtaining sample data in the critical region even though the null hypothesis is true.

Null hypothesis, aka H0, states that in the general population there is no change, no difference, or no relationship.

That it will have no effect on the dependent variable for the population.

What influences the power of your test?

The bigger your effect size, the more power your test has. ~power is the ability of your test to detect treatment effects if one exists. ~Larger sample sizes produce greater power. ~Using a 1 tailed test increases power.

Critical regions consist of the extreme sample outcomes that are "very unlikely".

The boundaries of critical regions are determined by the probability set by the alpha level.

The Alpha level is a small probability value that defines the concept of "very unlikely".

The critical region consists of outcomes that are very unlikely to occur if the Null hypothesis is true, where "very unlikely" is defined by the alpha level.

In a directional hypothesis test, it is a 1 tailed test!

The statistical hypothesis specify either an increase or a decrease in the population mean score. -They make a statement about the direction of the effect. -it is either positive (+) which is an increase, or negative (-) which is a decrease.

Directional hypothesis test, or a one-tailed test

The statistical hypothesis, H0 and H1, specify either an increase or a decrease in the population mean. That is, they make a statement about the direction of the effect.

The smaller the standard deviation, the smaller the standard error would be which means

The z area would be bigger & the critical areas would shrink. <--------->

The larger the standard deviation, the larger the standard error is which means

The z area would be smaller & the critical areas would be larger. >----<

Cohen's d = the mean difference divided by the standard deviation, or, M-u divided by sigma. Use this formula when there's no sample size, or n. So, Cohen's d is not affected by sample size.

This calculates how much influence the treatment has on the dependent variable. ~aka measure of effect formulas. ~d=0.2 is a small effect. ~d=0.5 is a medium effect. ~d=0.8 is a large effect.

Type 1 error

Treatment wasn't actually responsible for the effect as we concluded. -making the Alpha level smaller is more stringent & more restricted area for error-->Tightened up! -This reduces the risk of error from 5% to 1%.

There are 2 types of errors in decisions about rejecting or accepting the null; Type 1 & type 2 errors

Type 1 errors occur when a researcher rejects a null hypothesis that is actually true. Meaning that the the treatment was concluded as having an effect when, in fact, it did not have an effect. Type 2 error is when a researcher fails to reject a null hypothesis that is false; the test failed to detect a real treatment effect.

A smaller variance =

a smaller error which = a bigger z-score.

The size of difference between sample mean & original population mean

appears in the numerator of the z-score.

Hypothesis testing is an inferential process;

it is possible to make a mistake in your conclusion.

The *power* of a test is the probability that the test will correctly reject a false null hypothesis

meaning, it will detect a treatment effect if one exists.

Type 2 errors

missing a treatment effect that actually was there. -5% risk of error is traditionally used for this reason. --to make sure treatment exists, you replicate findings in several different areas.

~Reducing the alpha level, or making the test more stringent,

reduces power.

The 1.96 is a larger area of error in the 2 tailed test,

than a 1.65 z-crit in a 1 tailed test.

Anytime you have to choose between 2 z-score values *Always* choose

the larger to be more conservative.

You are *Always hypothesizing about the population*;

what would happen if we gave _(treatment)_ to the entire population? There are 4 steps: