Three t-tests. The t statistic. The main limitation of z. Using a t statistic. What do we have? What do we need? The single sample t-test

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1 The t statistic Three t-tests The single sample t-test A modified z test Also used when comparing two groups The paired sample t-test The independent sample t-test These are covered in the next lectures The main limitation of z Relies on knowing the population standard deviation In most cases, we do not know much about the population The t statistic Uses the sample variance as an estimate of the population variance It s the same with standard deviation Using a t statistic Usually, I curve my exams so that a 70 (approximately) is the mean These are five scores from a recent exam after the curve Are these significantly different than my usual post-curve scores? What do we have? What do we need? Population mean? Presented as 70 Population SD? Not given We must use the sample SD Thus we are going to use a t

2 Calculating the sample statistics The sample mean: 74 The standard deviation: Next we need the standard error s m = s/sqrt(n) s m = 17.04/2.23 = 7.6 Notice that N-1 is used for the standard deviation, but not used for standard error Formula for single sample t test t = (74-70) / 7.6 =.52 M µ s M M Now what? Use a t table df = 4 for this example Critical value is for p= is nowhere near this My sample is not statistically different than usual What is df? Degrees of freedom The number of scores that are free to vary when estimating a population parameter from a sample Let s say we have two scores in a group and we know the mean of the group is 10, and one of the scores is 8. The other score must be 12 One of the two scores is NOT free to vary The other is One degree of freedom here Baseball players Let s say you have a team of 9 players There are 9 positions on the baseball field How many of those players are free to vary? That is how many of them have options for where they can play? Only 8- why? Once the first eight are set, the ninth one only can take the last position

3 Do you notice the pattern? Two scores, one can vary 9 positions, 8 can vary Degrees of freedom for a single sample is N-1 OR df = N-1 EXAMPLE Researchers noticed that women (N = 20) from a certain area of the United States have an average height of 68 inches (SD = 4.5). It is believed that the average height for a woman is 65 inches (σ = 3.5). Researchers are trying to decide whether they should investigate the environmental factors surrounding this particular area that may be affecting the height of their women. Is this a significant difference or not? Go through your 6 steps 1. Identify the population and comparison distribution (your groups) 2. State the null and research hypotheses 3. Determine the characteristics of the comparison distributions 4. Determine the critical values or cutoffs 5. Calculate the test statistic Which statistics should you use here? You should use a z test Because you have the population SD Perform a t test on this data as well 6. Make a decision Steps 1 and 2 Population: Women in general Sample: women in a certain area of the US Null: There is no difference between the heights of women in this certain area of the US and the heights of women in general Experimental: Women in the certain area are taller than women in general Steps 3 and 4 Remember that we are comparing means These stats were given to us mainly: µ M = µ = 65 σ M = σ / sqrt(n) = 3.5/ sqrt (20) =.783 Critical values? One tailed hypothesis: z = 1.65 Step 5 With a z: z = (68-65) /.783 z = 3 /.783 = 3.83

4 Step 6: the decision z = 3.83 > 1.65 We will reject the null hypothesis There is a significant difference between the heights of women in this certain area and the heights of women in general Scientists should take a closer look at this area What if we used a t statistic? This would be necessary if a population standard deviation was NOT given to you You would use the sample SD as an estimate of the population σ SD = 4.5 Instead of finding σ M we will use s M = SD/sqrt (N) s M = 4.5 / sqrt (20) = 1.01 What about a critical value? We need to know df first Df = N-1 = 19 Critical value = t = for one tailed (α =.05) Step 5: With a t: t = (68-65) / 1.01 = Step 6: the decision > (critical value) We will reject the null as well using a t statistic here The difference between the heights of women in this area and women in general is significant What is the difference between what you found? Compare your critical values z was lower why? Compare your overall statistics t was lower why?

5 Effect size Calculate effect size d = (M - µ) / σ (68 65) / 3.5 =.86 This is a large effect size What if we didn t have the population SD? d = (M - µ) / s = (68 65) / 4.5 =.67 Thought piece Compare the z and t Which is more likely to give you a difference? Which is the more conservative test? That is, which test requires a larger difference to give significance?