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## Ch.9: Introduction to the t - statistic

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**Ch.9: Introduction to the t - statistic**First of 3 chapters (9, 10, & 11)**Chapter Overview**• Problem with using the z statistic in an hypothesis test • Introduction to the t-statistic • The sampling distribution of t values (the t distribution) • Hypothesis testing using the t statistic • Assumptions of the t-test. • Versatility of the t-test**Problem with using the z statistic in an hypothesis test**• Often, the population standard deviation is not known. • So we are not able to calculate the standard error of the mean • Standard error of the mean formula given**Introduction to the t-statistic**• When ơ is unknown, we use sample standard deviation (s) to estimate the population standard deviation (ơ) • Estimated standard error of the mean Sm = s/√n • t = M - µ Sm**The t distribution**• The t distribution = the sampling distribution of t values • How distribution of t can be derived manually. • Shape of t distribution depends on df where df = n-1 • Separate t distribution for each df (each sample size) • Reviewed df • The larger the sample size (n) the closer s is to ơ and the closer the t distribution is to the normal distribution (see Fig. 9.1, next slide)**Figure 9-1 Distributions of the t statistic for different**values of degrees of freedom are compared to a normal z-score distribution. Like the normal distribution, t distributions are bell-shaped and symmetrical and have a mean of zero. However, t distributions have more variability, indicated by the flatter and more spread-out shape. The larger the value of df is, the more closely the t distribution approximates a normal distribution.**Determining Proportions and Probabilities with the t-**distribution • To determine the probability of obtaining a t value we use the t distribution table (Table B.2, p. 693) • Example for t required for 2-tailed test, with df = 3 and alpha level of .05 (next slide) t required = -----------------------**Table 9-1 (p. 285)A portion of the t-distribution table.**The numbers in the table are the values of t that separate the tail from the main body of the distribution. Proportions for one or two tails are listed at the top of the table, and df values for t are listed in the first column.**Hypothesis testing using the t statistic**• Purpose of H: testing is to determine whether or not a treatment has an effect (if a treatment effect occurred) • t = M - µ = observed difference Sm difference expected by chance (sampling error) • Considered two major determinants of whether or not Ho is rejected based on above formula. • 1. ------------------------------------- • 2. ------------------------------------- • ------------------------------------------------- ---------------------------------------------------------**An Example of Hypothesis testing using the t statistic**• Will moth-eating birds avoid patterns that look like eyes (some moths have this pattern on their wings). • Method for above study presented • 5 steps in H: testing used to answer question.**Step 1: State the Hs: (null and alternative) and select**alpha level. Ho: H1: Select standard alpha level = .05**Step 2: Set the criteria for a decision**• Determine the critical region for rejecting Ho of no preference. • Since ơ is unknown, we must use s and the t distribution • For alpha = .05 and df = n-1 a t > ------- or • t < -------- identifies the critical region • (good idea to graph this)**Step 3: Collect sample data and calculate sample statistics**• Results: For sample of n=9 birds, the mean time spent in the plain side = 36 min. M = 36 • Sample variance. Not given raw data, but told that SS = 72 • Calculate sample variance and standard deviation • s2 = SS /df • s = √s • Calculate estimated standard error of the mean (Sm) • = ------------------------------ • For sample of n=9 birds, the mean time spent in the plain side = 36 min • (M = 36) • Calculate value of t statistic t = M - µ Sm t = -----------------**Step 4: Make a statistical decision**• Decision is about whether to reject Ho or fail to reject Ho. • What is the correct decision? • ----------------------------------**Concerns about hypothesis testing: Measuring effect size**• ----------------------------------------------- • ----------------------------------------------- • We can have a statistically significant treatment effect that is not practically significant (see Example 8.5) • Small but significant treatment effect**Measuring Effect Size**• We usually measure effect size when we report a significant result. • Cohen’s d = mean difference / standard deviation • Cohen’s d for z test: ------------------------------**Measuring effect size with 1 sample t - test**• Cohen’s d = mean difference (between M and u) Standard deviation (for the sample) • Cohen suggest criterion for evaluating effect size • -------------------------------------------- • -------------------------------------------- • -------------------------------------------- • Second method (calculate r2)**Measuring effect size with r2**• r2 = ------------------------------(remember from correlation) • For 1 sample t – test: • r2 = t2 t2 +df • Criteria for interpreting the value of r2 as proposed by Cohen -------------------------------------------------------**Step 5: Behavioural conclusion**• State what the statistical test indicated about research question (in APA style) • Examples given for eye spot pattern experiment. • --------------------------------------------------**Assumptions of the t-test**• The values in the sample must consist of independent observations (as in 1 sample z-test) • The population sampled should be normally distributed (as in 1 sample z test) • Can be violated, particularly if n≥30 • A 3rd assumption – not in text – was presented: • --------------------------------**Versatility of the t-test**• We do not need to know the population standard deviation • Also, we can use the t-test in some situations in which we don’t know the population mean. • That is, population mean is provided by Ho (as in worked example)