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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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**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 • H0: m1 - m2 = 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 (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**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)

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