1 Hypothesis Tests with the t Statistic (based on Gravetter & Wallnau) Review: Goal of hypothesis test: Is the obtained result significantly greater than we would expect by chance. 1. Sample mean approximates the population mean. 2. Standard error gives us an idea how well the sample mean approximates the population mean. It tells us how much difference between the sample mean and the population mean we might expect by chance. 3. We compare the sample mean with the hypothesized population mean with the z score test. Problem: We usually do not know the population standard deviation. Answer: Use the t statistic. 2 We will use the sample variability when the population value is not known. - See formulas in Chapter for the following Sample variance Sample standard deviation Estimated standard error “Estimated standard error (sM) is used as an estimate of the real standard error when the population standard deviation is unknown. It is computed from the sample variance or sample standard deviation and provides an estimate of the standard distance between a sample mean and the population mean” (p. 282). t = sample mean minus population mean divided by the standard error t statistic test hypotheses about an unknown population mean when the population standard deviation is unknown. Degrees of freedom “The larger the sample [and the greater the value of df], the better the sample represents the population.” Using the t distribution table 3 - See t distribution table in the back of the book Hypothesis tests with the t statistic With hypothesis testing, we begin with a population with an unknown mean and unknown variance, and a population that received a treatment. Goal: Use a sample from the “treated” population to determine if the treatment has any effect. t statistic: Numerator: Difference between sample data (sample mean) and population hypothesis (population mean). Denominator: How much difference is expected by chance (error). When the obtained difference between the sample data and the hypothesis (numerator) is greater than chance (denominator), we obtain a large t. Then we conclude data are not consistent with hypothesis, and we reject H O. Steps 4 1. State hypotheses (HO and HA) and set alpha level. 2. Determine df. Then use t distribution table (Appendix B) to locate critical region (and get the critical value). 3. Collect data. 4. Analyze data. Decide to retain or reject HO by comparing the calculated value with the critical value. Se Example 9.1 - n = 16 birds - Birds can roam in a box with two chambers: - One chamber has plain walls. - Other chamber has two large eye-spot patterns. - Birds tested one at a time in the box, for 60 minutes. - Amount of time spent in plain chamber was recorded. - Birds spend an average of 39 minutes in plain side, SS = 540. - Did the eye-spot pattern have an effect on behavior? 1. State hypotheses and set alpha. HO = Population mean, plain side is equal to 30 min HA = Population mean plain side is NOT equal to 30 min 5 p = .05 2. Locate critical region. df = n – 1 16 – 1 = 15 Look at t distribution table. critical region value: +/- 2.131 3a. Calculate test statistic. s2 = SS/df = 540/15 = 36 3b. Use sample variance (s2) to compute estimated standard error. sM = SQUARE ROOT OF s2/n = SQRT 36/16 = SQRT 2.25 = 1.50 3c. Calculate t statistic from sample data. t = sample mean-population mean/standard error = 39-30/1.50 = 9/1.50 = 6.00 ** 6.00 IS THE OBTAIN VALUE, ALSO KNOWN AS 6 THE CALCULATED VALUE 4. Make decision regarding HO. Measuring effect size for the t statistic mean difference Cohen’s d = _______________________ sample standard deviation Example 9.2 sample standard deviation S = SQRT SS/df = SQRT 540/15 = sqrt 36 = 6 Cohen’s d = 39-30/6 = 9.6 = 1.5 ** COMPARE 1.5 WITH TABLE 8.2 Sample Variance and Hypothesis Testing High variance means there are big differences from one score to another, which makes it difficult to see any consistent trends or patterns in the data. 7 ** High variance indicates noise and confusion in the data and makes it difficult to see what’s going on. ** When sample variance is large, the standard error also tends to be large. To obtain statistical significance, you must demonstrate that you results are greater than chance, where “chance” is measured by the standard error. When sample variance is large it is simply more difficult to obtain a statistically significant outcome.
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