Overview of Lecture Parametric Analysis is used for Single Sample Experiments Between Group Experiments Within Subject Experiments Homogeneity of Variance is important for Two Level Tests The statistic we use for all the above is the t-test

Single Sample Experiments These kind of inference are usually done under three conditions: The sample is relatively small in size The population is normally distributed (or assumed to be normal distributed) The population standard deviation, s , is unknown. The parameter we can examine are: The population mean The population variance

Single Sample Experiments When the population standard deviation is unknown, and The sample size is small, and The population is assumed to be normally distributed Then the sampling distribution of the statistic is not normal so we do not have a distribution against which we can compare.

Student’s t W.S. Gosset, writing as “Student”, developed the sampling distribution based on the statistic The sampling distribution of this statistic is known as Student's t distribution The t statistic has a family of sampling distributions that vary with respect to the degrees of freedom associated with the sample estimates used. t is associated with a sampling distribution that has N-1 degrees of freedom

One Sample t-test We can test if a sample mean is significantly different from a population mean using: The null hypothesis states that the difference between the sample mean and the population mean is zero The alternative hypothesis states that the difference between the sample mean and the population mean is greater than zero

One Sample t-test: Example We have measured the social desirability scores of a group of people. The population mean is 3 We are testing whether the sample mean is different to the population mean using:

One Sample t-test: Example First we have to calculate: The estimate of the population standard error of the mean:

One Sample t-test: Example So, t=2.049 is the single sample t associated with this data. We find a critical value for the t statistic by looking it up in tables with the appropriate degrees of freedom The critical value of the t statistic is equal to 2.262 for a two-tailed test at p<0.05. If tobserved  tcritical then reject the null hypothesis Since the observed value (2.049) doesn't exceed the critical value (2.262), we fail to reject the null hypothesis

One Sample t-test: Example When reporting a result we include four items The comparison made The statistical test used The degrees of freedom The significance level For this example: This sample has a mean social desirability score of 3.75 which is not significantly different from the population mean of 3 (t7=2.049, p<0.05)

Parametric Analysis of Two Condition Experiments When comparing the central tendency of two or more sets of scores we make a particular assumption about the variance: The variances of the two groups of scores are equal. This is known as the homogeneity of variance assumption

Homogeneity of Variance In the between groups design we use the test A critical value of F is looked up using N-1 degrees of freedom for both the numerator and denominator of the F ratio If the critical value in the tables exceeds the observed value then we reject the null hypothesis of no difference between the variances

Homogeneity of Variance With within subject designs alternative equation is use. Where

Between Group t-test Both z scores and t tests have relied upon standardization using the following We can use this again to derive the distribution for differences between the means of two independent samples

Between Groups t-test If the null hypothesis is true then the difference between the population means will be equal to zero since In other words, under the null hypothesis is an accurate representation of the difference between two means If this statistic is significantly greater than zero then we can reject the null hypothesis

Between Groups t-test The standard deviation of the sampling distribution of the mean differences is estimated using the statistic This formula simplifies to when the sample sizes are the same

Between Groups t-test A Doctor is interested in looking at the effectiveness of a drug that reduces pulse rates after exercise One group of subjects are given a placebo Another group of subjects are given the drug The pulse rates are measured after 3 minutes resting time

Between Groups t-test For this data and Therefore

Between Groups t-test Looking up t with 17 ( n1+n2-2 ) degrees of freedom the critical value is 2.110 at p<0.05 The observed value is 1.91 Since the observed value is not greater than the critical value we cannot reject the null hypothesis. For this data, there is no significant difference between the mean pulse rates of the placebo and drug groups.

Between Groups t-test When reporting a result we include four items The comparison made The statistical test used The degrees of freedom The significance level For this example: This the mean pulse rate of the drug group (91.11) is not significantly different to the mean pulse rate of the placebo group (107)with t17=1.91, p>0.05)

Repeated Measures t test The repeated measures t test is also known as the within subjects t test the matched subjects t test the related samples t test It is used when: there are two sets of scores provided by one group of subjects (a within subjects or repeated measures design) there are two set of scores provided by two groups of matched subjects

Repeated Measures t test The rationale of the matched t-test is only moderately different from the independent groups t-test. It begins by assuming that the relevant null hypothesis is that there is no difference between the scores that each individual subject in an experiment provides. If we adapt the single sample version of student's t to look at difference scores becomes

Repeated Measures t test The null hypothesis for this test is that the mean of the difference scores will be equal to zero The alternative hypothesis is that the mean of the difference scores will be greater than zero Under the null hypothesis the population mean of the difference scores is equal to zero therefore appropriate statistic for this test is: with N-1 degrees of freedom

Repeated Measures t test In this case the standard error of the mean difference scores is given by where is the estimate of the population standard deviation of the difference scores

Repeated Measures t test In this experiment subjects have be asked to judge the speed of two drivers. They have been told that one of the drivers is irresponsible and the other responsible

Repeated Measures t test For this data and So

Repeated Measures t test Looking up t with 7 ( N-1 ) degrees of freedom the critical value is 2.365 at p<0.05 The observed value is 5.196 Since the observed value is greater than the critical value we can reject the null hypothesis. For this data, there is a significant difference between the estimated speeds of the responsible and irresponsible drivers

Repeated Measures t tests When reporting a result we include four items The comparison made The statistical test used The degrees of freedom The significance level For this example: The mean rating of speed of the irresponsible driver (40.375) is significantly different to the mean rating of the speed of the responsible driver (32.625) with t7=5.196, p<0.05)