QUESTION 7 Questions 6 - 10 concern the following idea: The CEO of a company is trying to decide between two different suppliers of recycled aluminium (Alumania and Beautinium), she is not sure which one is better so is testing for a difference in either direction. She has decided to go with the supplier that has the smallest average impurities. 35 samples are obtained from each supplier and in each sample the percentage of impurity is measured. The results were analyzed in R via a 2-sample t-test with a 2-sided alternative hypothesis, and some of the output is summarized below (assume ? = 0.05 where necessary). mean sd n Alumania 3.67 3.34 35 Beautinium 4.01 3.74 35 Two Sample t-test data: impurity by supplier t = -0.413, df = 68, p-value = 0.6895 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -2.03 1.35 Levene's Test for Homogeneity of Variance (center = "median") Df F value Pr(>F) group 1 0.1977 0.658 68 Let ?1 and ?2 denote the mean amount of impurities from the suppliers “Alumania” and “Beautinium” respectively, and ?1^2 and ?2^2 be the respective variances. What is the pooled estimate of the standard deviation (correct to 3dp)?
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A standardized test has a scale that ranges from 3 to 45. A new type of review course for the test was developed by a training company. The accompanying table shows the scores for nine students before and after taking the review course. Complete parts (a) through (d) below: Perform a hypothesis test using a = 0.01 to determine if the average test score is higher for the students after the review course when compared with before the course. Let μd be the population mean of matched-pair differences for the score before the course minus the score after the course. State the null and alternative hypotheses: Choose the correct answer below: Ho: μd < 0 H1: μd > 0 Ho: μd = 0 H1: μd ≠0 Ho: μd < 0 H1: μd ≥ 0 Ho: μd > 0 H1: μd ≤ 0 Ho: μd = 0 H1: μd ≠0 b. Calculate the appropriate test statistic and interpret the results of the hypothesis test using a = 0.01. The test statistic is _________. (Round to two decimal places as needed.) The critical value(s) is(are) _________. (Use a comma to separate answers as needed. Round to two decimal places as needed.) Interpret the results of the hypothesis test: Since the test statistic after the review course _________ the critical value(s) _________, we fail to reject the null hypothesis. There is no evidence that the mean score is higher.
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Question 4 (10) a) Two different analytical tests can be used to determine the impurity level in steel alloys. Eight specimens are tested using both procedures, and the results are shown below. Use the Minitab output provided to answer the following questions. (i) Perform a hypothesis test to determine whether the differences between the observations are normally distributed. State your hypotheses, the test statistic, a p-value, the decision made and your conclusion. (ii) Test whether the mean level of impurity for both of the types of alloy are the same or not. State your hypotheses, a p-value, the decision made and your conclusion. Also indicate which output you used to make your decision. Output 5.1: Two-Sample T-Test and CI: Test 1, Test 2 Two-sample T for Test 1 vs Test 2 N Mean StDev SE Mean Test 1 8 1.450 0.220 0.078 Test 2 8 1.662 0.277 0.098 Difference = mu (Test 1) - mu (Test 2) Estimate for difference: -0.212 95% CI for difference: (-0.483, 0.058) T-Test of difference = 0 (vs not =): T-Value = -1.70 P-Value = 0.114 DF = 13 Output 5.2: Probability Plot of Test1-Test2 Normal Mean -0.2125 StDev 0.1808 N 8 AD 0.400 P-Value 0.274 Output 5.3: Paired T-Test and CI: Test 1, Test 2 Paired T for Test 1 - Test 2 N Mean StDev SE Mean Test 1 8 1.4500 0.2204 0.0779 Test 2 8 1.6625 0.2774 0.0981 Difference 8 -0.2125 0.1808 0.0639 95% CI for mean difference: (-0.3636, -0.0614) T-Test of mean difference = 0 (vs not = 0): T-Value = -3.32 P-Value = 0.01
An airline is trying two new boarding procedures, Option 1 and Option 2, to load passengers onto their Long Beach (LGB) to San Francisco (SFO) flights. Since Option 1 has more automation, the airline suspects that the mean Option 1 loading time is less than the mean Option 2 loading time. To see if this is true, the airline selects a random sample of 230 flights from LGB to SFO using Option 1 and records their loading times. The sample mean is found to be 17.7 minutes with a sample standard deviation of 5.2 minutes. They also select an independent random sample of 295 flights from LGB and SFO using Option 2 and record their loading times. The sample mean is found to be 18.3 minutes with a sample standard deviation of 4.1 minutes. Since the sample sizes are quite large, it is assumed that the population standard deviation of the loading times using Option 1 and the loading times using Option 2 can be estimated to be the sample standard deviation values given above. At the 0.05 level of significance, is there sufficient evidence to support the claim that the mean Option 1 loading time ÎĽ1 is less than the mean Option 2 loading time ÎĽ2 for the airline's flight from LGB to SFO? Perform a one-tailed test. 1. State the null hypothesis H0 and the alternative hypothesis H1? 2. Determine the type of test statistic to use? 3. Find the value of the test statistic? 4. Find the critical value at the 0.05 level of significance? 5. Can we support the claim that the mean Option 1 loading time is less than the mean Option 2 loading time for the airline's flights from LGB to SFO?
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