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Notes7 - 1 The t-test Introduction to Using Statistics for...

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1 The t -test Introduction to Using Statistics for Hypothesis Testing ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Overview of the t -test • The t -test is used to help make decisions about population values. We’ll start with how to do it, and come back to why later. There are two main forms of the t -test, one for a single sample and one for two samples. The one sample t -test is used to test whether a population has a specific mean value, e.g., whether the USF mean SAT-V is greater than 500. The two sample t -test is used to test whether population means are equal, e.g., do training and control groups have the same mean. ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Review of the Confidence Interval 95%CI = The confidence interval is the mean plus or minus a critical value of t times the standard error of the mean. The standard error of the mean is The standard error is just the standard deviation divided by the square root of N. The standard deviation is: X S t X 05 . ± N SD S X = 1 ) ( 2 = N X X SD ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________
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2 One-sample t -test We can use a confidence interval to “test” or decide whether a population mean has a given value. For example, suppose we want to test whether the mean height of women at USF is less than 68 inches. Suppose we randomly sample 50 women students at USF. We find that their mean height is 63.05 inches. The SD of height in the sample is 5.75 inches. Then we find the standard error of the mean by dividing SD by sqrt(N) = 5.75/sqrt(50) = .81. The critical value of t with (50-1) df is 2.01(find this in a t -table). Our confidence interval is, therefore, 63.05 plus/minus 1.63. See the graph. ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ One-sample t -test Example 80 70 60 50 Height in Inches = 63.05 =5.75 Pop Mean = 68 .01 Histogram of Sample Height Take a sample, set a confidence interval around the sample mean. Does the interval contain the hypothesized value? ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ One-sample t -test Example 70 Height in Inches The sample mean is roughly six standard deviations (St. Errors) from the hypothesized population mean. If the population mean is really 68 inches, it is very, very unlikely that we would find a sample with a mean as small as 63.05 inches. ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________
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3 Second Example Mean height = 66 in. (hypoth = 68 in.) SD height = 4 in. N= 25 women at USF • t .05 = 2.06 Standard error = ? CI = ? Test value of t = ? • Result? 8 . 25 4 = = = N SD S X 648 . 67 352 . 64 648 . 1 66 ) 8 (. 06 . 2 66 to CI ± = ± = 5 . 2 8 . 68 66 = = = X S X t µ Critical value of t is 2.06. Sample was not drawn from population with mean of 68 inches. ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Six Steps for Significance Tests 1 Set α (alpha or p level) • 2 State null and alternative hypotheses • 3 Calculate test statistic • 4 Determine critical value • 5 State decision rule • 6 State conclusion ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Example of Six Steps (1) 1. Set α . Alpha or p level is the probability of a Type I error (I will explain this later). By convention, alpha is typically set at .05 or .01. This tells you where to look for critical values in tables (like z , t and F ).
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