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Paired Differences ( Dependent ) Worksheet Answers 1. For a random sample of 20 data pairs, the sample mean of the difference was πΜ = 2. The sample standard deviation π π = 5. Assume that the distribution of the differences is mound β shaped and symmetric. At the 1% level of significance, test the claim that the population mean of the difference is positive. π»0 βΆ ππ = 0 π‘= πΜ βπ π π = π»1 βΆ ππ > 0 (2)(β20) 5 = 13.42 5 πΌ = 0.01 = 2.684 with π. π. = 19 π β π£πππ’π of π‘ = 2.684 is between 0.010 and 0.005 Since π β π£πππ’π β€ 0.01, we reject π»0 2. Are Americaβs top chief executive officers ( CEOs ) really worth all that money ? One way to answer this question is to look at row B, the annual company percentage increase in revenue, versus row A, the CEOβs annual percentage salary increase in that same company. Do the data indicate that the population mean percentage increase in corporate revenue ( row B ) is different from the population mean percentage increase in CEO salary? Use πΌ = 0.05 B : percentage increase for company A : percentage increase for CEO 24 23 25 18 6 4 21 37 21 25 20 14 -4 19 15 30 π©βπ¨ 3 -2 5 4 10 -15 6 7 Using a calculator : πΜ = 2.25 , π π = 7.78 , π = 8 , π. π. = 7 π»0 βΆ ππ = 0 π‘= πΜ βπ π π = π»1 βΆ ππ β 0 (2.25)(β8) 7.78 = 6.36 7.78 = 0.817 πΌ = 0.05 ( two β tailed ) with π. π. = 7 π β π£πππ’π of π‘ = 0.817 is between 0.250 and 0.500 Since π β π£πππ’π > 0.05, we fail to reject π»0 3. Do professional golfers play better in their first round? Let row B represent the score in the fourth ( and final ) round, and let row A represent the score in the first round of a professional golf tournament. A random sample of finalists in the British Open gave the following data for their first and last rounds of the tournament. Do the data indicate that the population mean score on the last round is higher than that on the first round ? Use a 5% level of significance. B : Last A : First π©βπ¨ 73 66 7 68 70 -2 73 64 9 71 71 0 71 65 6 72 71 1 Using a calculator : πΜ = 2 , π π = 4.5 , π = 9 , π. π. = 8 π»0 βΆ ππ = 0 π‘= πΜ βπ π π = π»1 βΆ ππ > 0 (2)(β9) 4.5 = 6 4.5 = 1.333 πΌ = 0.05 with π. π. = 8 π β π£πππ’π of π‘ = 1.333 is between 0.125 and 0.100 Since π β π£πππ’π > 0.05, we fail to reject π»0 68 71 -3 68 71 -3 74 71 3 4. In an effort to determine if rats perform certain tasks more quickly if offered a larger reward, the following experiment was performed. On day 1, a group of three rats, was given a reward of one food pellet each time they ran a maze. A second group of three rats was given a reward of five pellets each time they ran a maze. On day two, the groups were reversed. The average time in seconds for each rat to run the maze 30 times are show in the following table. Rat A B C D E F Time with one food pellet Time with five food pellets π©βπ¨ 3.6 3.0 0.6 4.2 3.7 0.5 2.9 3.0 -0.1 3.1 3.3 -0.2 3.5 2.8 0.7 3.9 3.0 0.9 Do these data indicate that rats receiving larger rewards tend to run the maze in less time? Use a 5% level of significance Using a calculator : πΜ = 0.4 , π π = 0.45 , π = 6 , π. π. = 5 π»0 βΆ ππ = 0 π‘= πΜ βπ π π = (0.4)(β6) 0.45 π»1 βΆ ππ > 0 = 0.98 0.45 = 2.178 πΌ = 0.05 with π. π. = 5 π β π£πππ’π of π‘ = 2.718 is between 0.025 and 0.010 Since π β π£πππ’π β€ 0.05, we reject π»0