Statistics- ONE AND TWO SAMPLE T-TESTS Homework 7

ONE AND TWO SAMPLE T-TESTSHomework 7Please include this sheet as a coversheet, fill it out and staple to your homework.NAME: __________________________________________________Class Time (OR “ONLINE”): _________________________________Recitation Day/Time or TA (blank if online): _____________________Grading:Ex 1Ex 2Ex 3Ex 4Ex 5Effort/neatness8881212TOTAL2Page 1 of 71. t-test of mean problem by hand The agricultural revolution in the 19th and 20th centuries is one ofthe socially significant problems that spurred the development of statistics. One important exampleis the discovery of the t-distribution.William S. Gosset (pictured at right) was an Oxford-educated mathematician,employed by the Guinness brewing company of Dublin. Since breweriesuse lots of grain, they are naturally involved in agricultural research, so partof Gosset’s work involved experimenting with crop yields. He realized thatnew mathematical techniques for handling the resulting data were needed,and he worked on the problem with Karl Pearson of University College,London. The outcome of Gosset’s study was perhaps the most famouspaper in statistical literature, “The Probable Error of a Mean” (1908), whichintroduced the t-distribution.Since Gosset was contractually bound by Guinness, he published under the pseudonym,“Student”, hence the t-distribution is often referred to as Student’s t-distribution. Here, we willanalyze one of the actual datasets that led to his analysis.There is a reason to believe that drying seeds before planting them will increase the yield of thecrop. As part of his analysis, Gosset reported on the results of 11 trials in which crop yields wereobtained from ‘regular’ seeds as well as ‘dried’ seeds, and the change in yield (after drying theseeds) was recorded.Because few prior studies had been done, nothing was known about the standard deviation ofthe population of all possible trials.a. If drying the seeds before planting had no effect on crop yield one way or the other, whatshould be the expected overall change in yield due to drying the seeds? [No calculation isnecessary here.]b. The question of interest is whether drying the seeds before planting will increase the cropyield. Please write the formal hypotheses for the test (your hypotheses should be stated insymbols – with relation to a population parameter – not words).Change in Crop YieldMean33.72727273Standard Error19.95134578Median38Mode#N/AStandard Deviation 66.17112801Sample Variance4378.618182Kurtosis-1.357191316Skewness-0.141741297Range197Minimum-70Maximum127Sum371Count11Some summary statistics on the eleven data values are shown to theleft.c.From the output, report the value of the appropriate samplestatistic (use correct symbol to identify).d.To compute the standard deviation of the sampling distribtution(the standard error), you will need one other value. In this case,what is the symbol for the other value that you need, and what is itsmeaning?Page 2 of 7e. Graphically represent a bell-shaped distribution of the appropriate statistic assuming the NullHypothesis is true. Label the mean of the distribution with the appropriate value; then markthe value of the sample statistic in the appropriate location on the sketch, and shade theassociated ‘tail’ probability (in other words, draw your rejection region).f. The value that you described in (d) can be found on the descriptive statistics shown above.Do this, and then compute and report the standard error [show calculations].g. Notice that Excel also computed the same result you just obtained in part (f). Did the valuethat you calculated match what is listed in the output?h. Calculate and report the test statistic, along with its appropriate symbol. Show allcalculations.i. Use the table to find the critical value of the statistic at a significance level of 0.05 and stateyour rejection region in terms of this value. Use the appropriate notation for the criticalvalue.j. Decide and support your decision using the rejection region approach.k. Conclude in terms of the problem. You should state the meaning of your decision in part (j)in the context of the question regarding whether the experiment gives good evidence thatdrying seeds before planting does, or does not, increase the crop yield.Page 3 of 72. Companies that sell groceries over the internet are called e-grocers. Customers enter their orders,pay by credit card, and receive delivery by truck. A potential e-grocer analyzed the market anddetermined that the average order would have to exceed $85 if the e-grocer were to be profitable.To determine whether an e-grocery would be profitable in one large city, she offered the serviceand recorded the size of the order for a random sample of customers.a. State the null and alternative hypotheses to determine whether and e-grocer will beprofitable in this city.b. Choose the test statistic that will be used for this inference test, write the name of the testthat will be used.The following Microsoft Excel output shows the result of the statistical test (from ‘TestStatistics.xls’):Sample meanSample standard deviationSample sizeHypothesized meanAlpha89.2717.385850.05t StatP(T<=t) one-tailt Critical one-tailP(T<=t) two-tailt Critical two-tail2.280.01271.66320.02541.9886c. Using the information in the output above, create a sketch of this test. On your sketch, youshould have the following labeled with proper symbols and values: population mean,sample mean, statistic, significance level (shaded), p-value (shaded).d. Use the p-value approach to make a decision with respect to the hypotheses. Support yourdecision with results from the output.e. Write a 1-2 sentence conclusion in terms of the problem advising the potential e-grocer ofwhat she should do.Page 4 of 7[The following scenario/data will be used for problem 3]One factor in low productivity is the amount of time wasted by workers. Wasted time includes timespent cleaning up mistakes, waiting for more material and equipment, and performing any otheractivity not related to production. In a project designed to examine the problem, an operationsmanagement consultant took a survey of 200 workers in companies that were classified as successful(on the basis of their last annual profits) and another 200 workers from unsuccessful companies. Theamount of time (in hours) wasted during a standard 40-hour workweek was recorded for each worker.3. Do these data provide enough evidence at the 1% significance level to infer that the amount oftime wasted in unsuccessful firms exceeds that of successful ones?a. State the null and alternative hypotheses.b. Based on your conclusion from the statistical test that you carried out in problem 3, what testshould be used to test the hypotheses that you put forth in part (a) [state the full name of thestatistical test]?c. The sample data for the two groups was reported as follows:SuccessfulUnsuccessful5.027.80s1.393.09n200200Using the Test Statistics workbook in Excel (file is on Courseweb under “CourseDocuments”), conduct the test that you have listed in part (b). Attach your results.d. Using your results from part (c), make a decision with respect to your null and alternativehypotheses. Support your decisions with values from your output.e. List the conditions that must be met in order to use this test – how would you go abouttesting them?f. Write 2-3 sentences that conclude the test that you carried out in problems 3 and 4 thatcould be used by the operations-management consultant in his report to an unsuccessfulcompany.Page 5 of 74. Use computer software to test the hypotheses that you stated in Homework 6, part 6c. You maychoose the significance level at which you will carry out your test (make sure that it’s notedsomewhere). You should carry out the following steps (as we have practiced in class and on all ofthe hypothesis testing problems) and attach to your assignment:a. State hypothesesb. Choose the proper statistical test to be used and state what information helped you to cometo this decision. Check any conditions that you may have to meet in order to carry out yourtest (show any related output). Perform the test using Microsoft Excel (using the TestStatistics workbook) and attach your output.c. State your decision and support it with values from the output.d. Conclude in terms of your topic.5. Answer the following questions about the dataset that you’re analyzing this term:a. List the categorical (nominal or ordinal) variables that you have:b. Try to find a statement that someone has made about one of these categorical variablesthat you have listed in part (a). For example, if I chose to analyze student satisfaction withthe current textbook, I may use a statement made by the the publisher that “The majority ofbusiness statistics love this book.” If you cannot find a statement online, you can asksomeone about the topic (that is, ask the question “what do you think about the businessstatistics book you’re using? Do you (1) LOVE IT, (2) don’t care, (3) hate it”) and use thisas your value. Copy or write the statement below:c. Make a claim. What do you think about this claim? Do you think that the true value willactually be higher, lower, or different (that is, you’re not sure the direction of the difference)?d. Write null and alternative hypotheses putting your answers in parts a and b in the form of ahypothesis about the true population proportion p. [use appropriate symbols]Page 6 of 7



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