stats week 2

stats week 2

1. the t test

   • used when we have 1 IV with 2 levels - estimates whether the pop means under the 2 levels of iv are different (estimate is based on the difference between measured sample means)

2. variance

   • variance between iv levels = variance between the two separate sets of results - variance within IV levels = variance between each result in group A for example
   • what contributes to variance between IV levels? manipulation of IV, individual differences, experimental error (random error, constant error)
   • what contributes to variance within IV levels? individual differences, experimental error (random error)

3. sampling distribution of differences & t-test

   • null hypothesis states that there is no difference between the pop. means so under the null hypothesis, the sampling distribution of differences will have mean of 0 - the t-dist represents the distribution of sampled mean differences when the null hypothesis is true
     • the t-dist has a mean of 0 - the extent to which an individual sampled mean difference deviates from 0 can be expressed in SE units - (standard error because this is a sampling distribution) - we convert the difference between out sample means into a t-value by expressing the difference in SE units
       • so we need to convert the difference between our sampled means into a t-value - but first we need to know the SE or the mean difference - if we had all measured possible sample differences, we could calculate this - but we don't - so we can estimate the SE of mean differences based on the sample SDs and sample sizes

4. Independent t-test formula

   • t is a ratio - it reflects the difference between the sample means, expressed in standard error units - we can use the t-dist to determine the probability of measuring a t-value of the magnitude we have obtained if the null were true
     • t = variance between iv levels / variance within IV levels - t value close to 0 = small variance between IV levels relative to within - t value further from 0 - large variance between IV levels relative to within
       • critical values of t depend on degrees of freedom - degrees of freedom is the difference between the number of measurements made and the number of parameters estimated - i.e sample size - no. parameters - i.e if we want to estimate a population mean with sample size of 12, we have 11 df - the larger the df, the more reliable the estimate
         • df for independent t test - difference between number of measurements made and number of parameters estimates - we have 2 samples in an independent t test and we estimate one parameter for each sample (2 in total) - so n (total for both samples) - 2
           • using out t dist we can determine the prob of obtaining a t-value of a given magnitude when null is true - this is out p-vaue - (0.05 threshold)
       • assumptions for independent t-test - normal distribution, homogeneity of variance (variance in DV under each level of IV should be reasonably equivalent, sample sizes should be roughly equal, scores under each level should be independent

5. effect size

   • the magnitude of difference between 2 IV level means, expressed in SD units - i.e a standardised value expressing the difference between IV level mean - this allows us to compare the difference between means obtained with different studies
     • d = difference between sample means / mean SD
       • so we can say 'our means differ by 2.55 SD' - small effect size = 0.2, medium = 0.5, large = 0.8

6. paired t-test

   • similar but difference calculations - difference score calculated for each participant (as its RM) - still uses variance between IV levels / variance within IV levels
     • what causes variance is difference - variance between IV levels caused by manipulation of IV and experimental error (no ID because of RM design) - within = experimental error
       • assumptions - normal distribution, sample size roughly equal
         • df for paired t-test = 1 sample and 1 parameter so n total - 1