ass_4_solution

3. Two different brands of milk are randomly sampled, and the fat content in each bottle of milk is determined. Twenty-six bottles of Brand A milk yielded an average fat content of ̄ ? 1 = 25 grams with ? 2 1 = 4 , and thirty one bottles of Brand B yielded an average fat content of ̄ ? 2 = 25.8 grams with ? 2 2 = 7 (this question is worth 3 marks). Test the hypothesis that both brands have identical average fat content at the 5% level of significance. Let 𝜇 1 be the population mean fat content of Brand A milk and let 𝜇 2 be the population mean fat content of Brand B milk. 𝐻 0 ∶ 𝜇 1 = 𝜇 2 𝐻 1 ∶ 𝜇 1 ≠ 𝜇 2 ̄ ? 1 = 25 , ? 2 1 = 4 , 𝑛 1 = 26 ̄ ? 2 = 25.8 , ? 2 2 = 7 , 𝑛 2 = 31 ? 2 𝑝 = (𝑛 1 −1)𝑠 2 1 +(𝑛 2 −1)𝑠 2 2 𝑛 1 +𝑛 2 −2 = (26−1)×4+(31−1)×7 26+31−2 = 25×4+30×7 26+31−2 = 310 55 ≈ 5.64 ? 𝑝 = √ ? 2 𝑝 = √ 5.64 ≈ 2.37 ? = ̄ 𝑋 1 − ̄ 𝑋 2 𝑠 𝑝 √ 1/𝑛 1 +1/𝑛 2 = 25−25.8 2.37×√ 1/26+1/31 ≈ −1.27 ?? = 𝑛 1 + 𝑛 2 − 2 = 26 + 31 − 2 = 55 # p-value pt ( - 1.27 , df = 55 ) * 2 [1] 0.2094315 Using the two-sample t-test, we cannot reject the null hypothesis at 5% level of significance, ?(55) = −1.27 , 𝑝 = .209 . The fat content does not statistically significantly differ between the two brands of milk. 4. To compare two programs for training industrial workers to perform a skilled job, 20 workers are included in an experiment. Of these, 10 are selected at random and trained by method 1; the remaining 10 are trained by method 2. After completion of training, all the workers are subjected to a time-and-motion test that records the speed of performance of a skilled job. The following time, as measured in minutes, is obtained. Method Method 1 15 20 11 23 16 21 18 16 27 24 Method 2 23 31 13 19 23 17 28 26 25 28 4