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Test #3 Math 146 Name: Please, return these sheets only, writing your answers in the blank spaces. You may add sheets if you definitely run out of space, though. Show all your work: we are not interested in the result, but in the method you use to find it. In particular, when providing a "statistic" (like the mean, the standard deviation, and so on) indicate the formula you are using. When grouping data, as in finding a percentile, indicate (very briefly) what you are doing, and how come you got that value (e.g., "I had 100 sorted data, the 25th and 26th were 34 and 37, so I chose <fill in your choice - you know there is an element of arbitrariness here> as the 1st quartile" A special note on calculators: You obviously will need a calculator to produce some of the answers to the questions (unless you are unusually fast at doing calculations by hand). I expect you to have one with you. If, for whatever reason, you don't have one, please mention this fact here. You can still answer all questions, and when a calculation is impractical to do by hand, write the formula you would have keyed into your calculator, had you had one. As a simple, short example, if you were supposed to calculate the square of 4,827.19, you should write, in place of the result, 4817.19 2 , and, if this result was supposed to be used in a subsequent calculation, write the formula for that second calculation, inserting this expression where appropriate. As a simple, short example, if this result should then have been divided by the square of 35, you would write 4817.19 2 35 2 . About confidence and significance levels: choose your preferred level, both for the confidence interval and the test questions. It is appreciated if you can provide an estimate for the p – value of the tests (without a computer, it is unlikely you could pinpoint a precise value, since we are dealing with t tests), and consequently discuss whether there are levels that would cause you to reverse your conclusion (for example, in a 2-tailed test, a p – value of 0.021 would cause you to reject the null hypothesis at the 0.05 level, but not at the 0.01 level, or even the 0.04 level, for that matter) We will work on two data sets, separately, and together. Pay attention to which one(s) are referred to in the questions. Both are the results of computer simulations purported to be samplings from normal distributions. Data set 1 is listed, together with the usual summaries Data Set #1 0.36510 0.66417 0.86830 1.32485 0.37426 0.68198 0.91983 1.37512 0.48117 0.71663 0.93507 1.54369 0.59840 0.77177 1.02087 1.94499 0.64531 0.82731 1.26636 1.97105 Sum Count 19.2962 20 Sum of Squares 22.7967 Data set 2 is summarized only, reporting the sample mean, the sample variance, the sum of the data, and the count Data Set #2 Mean 0.79618 Sample Variance 0.02053 Sum 23.8855 Count 30 Recall: The sample standard deviation is the square root of the sample variance Descriptive Statistics: Calculate for data set #1 only • Sample mean, sample standard deviation, and sample variance • 1st Quartile, Median, and 3rd Quartile Inferential Statistics Confidence Interval for data set #1 Choosing a confidence level, use the previous data for a confidence interval for the “true” mean (the variance is unknown). Optional: If you have time left, you may also construct a confidence interval for the second data set. Testing for the mean for data set #2 Test the hypothesis (at a level of your choice) H0: μ = 0.75 H0: μ ≠ 0.75 Testing for equality of two means (clearly, this refers to both data sets) We now want to check if the two data sets could come from “populations” (i.e., probabilistic models) with the same “true” mean, assuming they are independent samples. Set up a test, without assuming that the variances are equal. Optional: If you have time left, you can repeat the test, assuming, instead, that the populations have equal variances.