What are their purposes? What kinds? T-Tests What are their purposes? What kinds?

Case I Research Does a particular sample belong to a particular population? Testing a null hypothesis of no difference between the sample mean and the hypothesized population mean is tested.

Case II Research 1) Provides data on whether two means are drawn from identical populations or from different populations 2) Draws subjects randomly from some population and measures their behavior under two conditions such as two occasions. Or subjects can be matched on some variable and then randomly assigned to an experimental and control group. Testing the null hypothesis of no treatment effect.

Single samples t-test Independent t-test Dependent t-test Case 1 Research Independent t-test Case 2 Research Dependent t-test

.Assumptions: The population from which the sample was drawn is normally distributed The sample or samples are randomly selected form the population When two samples are drawn, the samples have approximately equal variance. The variance of one group should not be more than twice as large as the variance of the other (Homogeneity of variance) Data must be parametric (based on interval or ratio measurement scale

For very small samples: The assumption of a normal population cannot be examined. If N >30 t-test can be used even if the distribution of scores in the parent pop is somewhat non-normal. If N is small , non-normality may give rise to error.

When one or more of the criteria are not met: then to avoid errors, a more conservative p value (ie p<.01 compared to p<.05) should be used

Note 1. If the population mean is known or assumed to be a certain value, and if the sample mean is not close enough to the population mean to fall within the limits sets by a selected level of confidence then:

The sample mean was not drawn from the population The sample was drawn from the population but was modified so that it is no longer representative of the population from which it was originally drawn.

We can show the same with two sets of data: If we draw two samples from the same population and the means differ by amounts larger than would be expected based on normal distributions, one of the following must be true:

One or both of the samples were not randomly drawn from the population Some factor has affected one or both samples causing them to deviate from the population from which they were originally drawn

When samples are used to estimate population parameters especially when the samples are small t distribution must be used to evaluate the t statistic. Critical Ratios = The Student’s t Distribution at the given p values

To interpret t for a sample: Must first find the degrees of freedom.

Degrees of freedom: The independent pieces of info on which a sample statistic is based. Often used for the number of such pieces of info.

Mean based on 40 scores is determined by all 40 scores – each scores provides an independent piece of info in the mean’s calculation. Generally, the number of df on which a sample statistic is based depends on the sample size (N) and on the number of sample statistics used in its calculation

Then: The t ratio is compared to the values for the two-tailed test from the Student’s t distribution for the appropriate df.

Next: When the t exceeds the value in the table for a given p level, we conclude the X was not drawn from µ

Is useful to see whether the techniques introduced by an experiment had an impact on the participants. If we know or estimate the population parameters and then draw a random sample and treat it in the manner that is expected to alter its mean value, we can determine the odds that the treatment had an effect by using a t-Test.

If t exceeds the critical ratios we can conclude that the treatment was effective because the odds are high that sample is no longer representative of the population from which is was drawn. The treatment has caused the sample to change so that it does not match the characteristics of the parent population.

When t is less than the critical ratio , the null hypothesis Ho is accepted : there is no reliable difference between X and µ

When t exceeds the critical ratio, Ho is rejected and H1 is accepted; some factor other than chance is operating on the sample mean.

Example 1. Suppose a random sample of 9 college students was drawn from university freshman at WSU. The mean score on the IQ test was 125 with a standard deviation of 10. Because the IQ was developed so that the mean equals 100 in the general population, the question arises as to whether freshman in this university have on the average higher IQ scores than the general population.

What case type of research is this?

Null & alternative hypotheses stated: Ho: µ = 100 No difference exists in freshman college students’ IQ compared to the population mean. H1: µ ≠ 100 A difference exists in freshman college students’ IQ compared to the population mean.

Formula: t observed = X - µ S x = X - µ S/ N = 125 – 100 10/ 9 = 7.50

Decision Rules Observed t value = 7.50 Next: reject or fail to reject the null on basis of t observed

Decision Rules cont. For a two – tail test For a one - tail test A. Reject Ho if t observed > t critical ( p/2, df) B. Reject Ho if t observed < t critical ( p/2, df) For a one - tail test A. Reject Ho if t observed > t critical ( p, df) B. Reject Ho if t observed < t critical ( p, df)

For a single sample t-test N – 1 df 9 – 1 = 8 df Find critical value of t in Students’ t distribution at p<.05

Ho: µ1 - µ2 = 0 H1: µ1 - µ2 = 0 i.e., Ho: µ1 = µ2 H1: µ1 = µ2 Independent Means Ho: µ1 - µ2 = 0 H1: µ1 - µ2 = 0 i.e., Ho: µ1 = µ2 H1: µ1 = µ2

Independent t-test: Ho: µc - µe = 0 H1: µc - µe < 0 Where: C = control group E = experimental group p<.01

Dependent Samples t-test: 1. One independent variable with two levels (two cells in the design) 2. The same group of subjects is observed under both treatment conditions or matched pairs of subjects are observed so that one member of the pair is observed under treatment 1 and the other is observed under treatment 2.

3. The levels of the independent variable may differ from one another either quantitatively or qualitatively.

Ho: µD =0 H1: µD = 0 p<.01