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# t-test

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 

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## t-test

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1. t-test Mechanics

2. 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

3. 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

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

5. 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 • H0: m1 - m2 = 0

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

7. 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

8. 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)

9. Formula • All of which leads to:

10. But... • If the null hypothesis is true:

11. t test • Reduces to:

12. Degrees of freedom • Across the 2 samples we have (n1-1) and (n2-1) degrees of freedom df = (n1-1) + (n2-1) = n1 + n2 - 2 *Refer again to p. 50 in Howell with regard to the interpretation of degrees of freedom

13. 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

14. Weighted average Which gives us: Final result is:

15. Pooled variance estimate • Before we had • Now we use our pooled variance estimate

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

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

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