Lab Section 9 Part 3 Variance Key (1)

STAT 3090 L AB S ECTION 9 P ART 3 F ALL 2023 H YPOTHESIS T EST V ARIANCE AND E RRORS N AME : Key O BJECTIVES : Upon completion of this lab, you should be able to:  State and verify the conditions necessary for a hypothesis test for variance/standard deviation  Locate the Rejection Region for a hypothesis test for variance/standard deviation using the chi squared distribution  Perform a Hypothesis Test for a variance/standard deviation  Calculate a P-value for a hypothesis test for a variance/standard deviation  Draw conclusions about the population based on the results of a sample  Describe a Type I and II error in context 1. N OODLE L ENGTHS M ACHINE #13 (45 pts) D IRECTIONS : For the following hypothesis test for variance , you should define your parameter of interest and follow the HATS steps to complete the hypothesis test. Address each step in your answer. Previously in STAT 3090 … As you may recall from Labs 8 and 9 (Proportions and Means) and JMP Test 2, one of the machines that makes the spaghetti noodles for Delectable Delights, machine #13, has been malfunctioning. The noodles have not been a consistent length. First the noodles were too long, and now they are too short. By this time the employees at Delectable Delights are, well, for lack of a better term, freaked out. (They have no idea that the Weasley boys have been using the machine to test their Too Long Spell and their Too Short Spell.) A repair service was called and machine #13 was recalibrated to produce noodles with lengths that are normally distributed and on average 252.5 mm long. In the JMP Quiz 2, you used a sample of noodles from Machine #13 to check whether the mean length is within spec. But the other requirement for noodle manufacture is that the lengths of the noodles should not vary too much—in particular, the standard deviation of noodle lengths should not exceed 2.3mm. In a sample of 65 randomly selected noodles, the standard deviation of noodle lengths was 2.55mm. Complete a hypothesis test to determine if there is evidence at the 0.05 level that machine #13’s output is too variable. Write up your result using the HATS procedure (address each of the steps in your answer). Use P-value approach or the rejection region approach with the appropriate table from your course notes. You may assume that the population of noodle lengths is approximately normally distributed. 1

STAT 3090 L AB S ECTION 9 P ART 3 F ALL 2023 H YPOTHESIS T EST V ARIANCE AND E RRORS Let σ 2 = the population variance of the length of spaghetti noodles from machine #13 H 0 : σ 2 = 5.29 H a : σ 2 > 5.29 Assumptions Step The sample is randomly selected as stated in the problem. The noodle lengths are approximately normally distributed as stated in the problem. Test Step χ 0 2 = 64 ( 2.55 2 ) 2.3 2 = 78.669 P-value approach Rejection Region Approach P-value = P ( χ 2 > 78.669 ) = 0.1026 RR: χ 0 2 > 90.531 (using 70 df) or > 79.082 (using 60df) 0.1026 > α = 0.05 78.669 < 79.082 or 90.531 Since the P-value is greater than α Since the test statistic is not in the Rejection Fail to reject H 0 Region, Fail to reject H 0 Summarize Step At the 5% significance level, we do not have sufficient evidence to conclude that machine #13 is out of spec with respect to variability. 2 - 5 points for checking the randomness assumption - 5 points for checking the normality assumption - 2 points for writing the formula - 3 points for substituting the correct values - 5 points for the correct test statistics - 5 points for writing the correct p-value or Rejection Region - 5 points for providing the correct decision about the null hypothesis with justification - 5 points for using the correct significance level - 5 points for providing “sufficient evidence” or an equivalent - 5 points for writing the conclusion in context of the problem