t-test Mechanics

Z-score If we know the population mean and standard deviation, for any value of X we can compute a z-score Z-score tells us how far above or below the mean a value is in terms of standard deviation

Z to t Most situations we do not know  However the sample standard deviation has properties that make it a very good estimate of the population value We can use our sample standard deviation to estimate the population standard deviation

t-test Which leads to: where And degrees of freedom (n-1)

Independent samples Consider the original case Now want to consider not just 1 mean but the difference between 2 means The ‘nil’ hypothesis, as before, states there will be no difference  H 0 :  1 -  2 = 0

Which leads to... Now statistic of interest is the difference score: Mean of the ‘sampling distribution of the differences between means’ is:

Variability Standard error of the difference between means Since there are two independent variables, variance of the difference between means equals sum of their variances

Same problem, same solution Usually we do not know population variance (standard deviation) Again use sample to estimate it Result is distributed as t (rather than z)

Formula All of which leads to:

But... If the null hypothesis is true:

t test Reduces to:

Degrees of freedom Across the 2 samples we have (n 1 -1) and (n 2 -1) degrees of freedom df = (n 1 -1) + (n 2 -1) = n 1 + n *Refer again to p. 50 in Howell with regard to the interpretation of degrees of freedom

Unequal sample sizes Assumption: independent samples t test requires samples come from populations with equal variances Two estimates of variance (one from each sample) Generate an overall estimate that reflects the fact that bigger samples offer better estimates Oftentimes the sample sizes will be unequal

Weighted average Which gives us: Final result is:

Pooled variance estimate Before we had Now we use our pooled variance estimate

Paired t test Where = Mean of difference scores = Standard deviation of the difference scores n = Number of difference scores (# pairs)

Degrees of Freedom Again we need to know the df df = n-1  n = number of difference scores (pairs)