1) For each of the following scenarios, define the null hypothesis (Ho) and the alternate hypothesis(H1):(a) The mean diameter of bolts produced by Machine 2 is tested to see if is different from thediameter of bolts produced by Machine 1(b) The standard deviation of bolts produced by Machine 1 is tested to see if it is greater thanthe standard deviation of bolts produced by Machine 2.(c) The proportion of bolts produced by Machine 2 that do not meet specification is 2% lessthan that of bolts produced by Machine 1.Note: For #1, in addition to listing the hypotheses on the cover sheet, include a shortexplanation of the reason(s) for choosing those hypotheses as part of your supplementarymaterials. 4) Your construction company has purchased two lots of southern yellow pine lumber, bothsupposedly with 4% moisture content. The mean perpendicular compression strength of thefirst lot is tested with a sample of n1 = 13 boards, yielding x = 14.8 MPa. A test of a sampleof n2 = 13 boards from the second lot, which arrives after several days of rain and is wet,results in x, = 13.7 MPa. The population standard deviations are known to be the same, atof = ož = o? = 0.5 MPa. Complete the following:(a) Conduct a hypothesis test with a = 0.01 to see if the mean perpendicular compressionstrengths of the two populations are the same or not. Your work should show all 7 steps ofthe hypothesis testing procedure. On the cover sheet, just write your conclusion. (i.e.,"There is/is not sufficient evidence at the a = 0.01 level of significance to conclude that themean perpendicular compression strengths..")(b) For the results of your hypothesis test in part (a), what is the p-value for this test?(c) What is the probability that you will incorrectly reject the null hypothesis? Provide thecorrect symbol and value in your answer on the answer sheet.(d) To calculate the probability that you will fail to reject an incorrect null hypothesis, supposethat the true difference in mean perpendicular compression strength between the two lotsis A= µ1 – µz = 0.5 MPa. Calculate this probability, and provide the correct symbol andvalue in your answer on the answer sheet.(e) Suppose that you want the reduce the probability of a Type II error to ß =0.10, so that= 0-1(0.10) = 1.28. If all other parts of the problem remain the same, what is theminimum number of samples that you will need?(f) Suppose you measure the sample standard deviations and find that s, = 0.51 MPa andS2 = 0.75 MPa, with sample sizes the same at 13 boards each. Conduct a hypothesis testwith a = 0.05 to see if the standard deviations of the two lots are the same or not. As in(a), show all 7 steps and write the full conclusion on your answer sheet.

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A First Course in Probability (10th Edition)

10th Edition

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Chapter1: Combinatorial Analysis

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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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