UIUC STAT 400 - Week 12 ‐ Discussion Questions

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Week 12 Discussion Questions STAT 400 Spring 2022 D Unger Table of Contents Exercise 1 Hypothesis Test for Error Bookmark not defined Exercise 2 Hypothesis Test for 1 Exercise 3 Hypothesis Test for 2 Exercise 4 Hypothesis Test for p 2 Exercise 5 Hypothesis Test for p 2 Exercise 1 Confidence Interval for The overall standard deviation of the diameters of a certain set of ball bearings is 0 005 mm The overall mean diameter of the ball bearings must be 4 300 mm A sample of 81 ball bearings had a sample mean diameter of 4 299 mm Is there a reason to believe that the actual overall mean diameter of the ball bearings is not 4 300 mm a Perform the appropriate hypothesis test using a 10 level of significance Step 1 the actual overall mean diameter of the ball bearings H0 4 300 vs H1 4 300 Step 2 cid 3364 the sample mean diameter of the ball bearings cid 3364 Normal 2 n with known We will use cid 3404 cid 3025 cid 3364 cid 2879 cid 3091 cid 3097 cid 3041 Solution Step 3 Step 4 Step 5 There is significant evidence to suggest that the actual overall mean diameter of the ball bearings is not 4 300 mm b Repeat the hypothesis test in part a but provide evidence in the form of an appropriate confidence interval instead Solution Solution Solution Since the 90 confidence interval of 4 29809 4 29991 does not include the null hypothesis claim of 4 300 then we reject H0 at the 0 10 level of significance c Repeat the hypothesis test in part a but provide evidence in the form of the test statistic and critical rejection region instead Since the test statistic value of z 1 80 1 645 then it lies in the rejection region and we reject H0 at the 0 10 level of significance d Repeat the hypothesis test in part a but this time using a 5 level of significance The data test statistic and p value are unchanged But now we have that 0 0718 0 05 Since p value then we fail to reject H0 at the 0 05 level of significance We conclude that there is not significant evidence to suggest that the actual overall mean diameter of the ball bearings is not 4 300 mm Exercise 2 Confidence Interval for A trucking firm believes that its mean weekly loss due to damaged shipments is at most 1800 Half a year 26 weeks of operation shows a sample mean weekly loss of 1921 54 with a sample standard deviation of 249 39 Perform the appropriate hypothesis test using an 0 10 level of significance Step 1 the overall mean weekly loss due to damaged shipments H0 1800 vs H1 1800 Step 2 cid 3364 the sample mean weekly loss due to damaged shipments cid 3364 Normal cid 3025 cid 3364 cid 2879 cid 3091 2 n with unknown We will use cid 3404 cid 3046 cid 3041 Solution Step 3 Step 4 Step 5 There is significant evidence to suggest that overall mean weekly loss due to damaged shipments is greater than 1800 Exercise 3 Confidence Interval for Metaltech Industries manufactures carbide drill tips used in drilling oil wells The life of a carbide drill tip is measured by how many feet can be drilled before the tip wears out Metaltech claims that under typical drilling conditions the life of a carbide tip follows a normal distribution with mean of at least 32 feet Suppose some customers disagree with Metaltech s claims and argue that Metaltech is overstating the mean i e the mean is actually less than 32 Metaltech agrees to examine a random sample of 25 carbide tips to test its claim against the customers claim If the Metaltech s claim is rejected Metaltech has agreed to give customers a price rebate on past purchases Suppose Metaltech decided to use a 5 level of significance and the observed sample mean is 30 5 feet with the sample variance 16 feet2 Perform the appropriate hypothesis test Solution Step 1 the population mean lifetime of a drill tip measured in distance feet H0 32 vs H1 32 Step 2 cid 3364 the sample mean lifetime of a drill tip measured in distance feet cid 3364 Normal 2 n with unknown We will use cid 3404 cid 3025 cid 3364 cid 2879 cid 3091 cid 3046 cid 3041 Step 3 Step 4 Step 5 There is significant evidence to suggest that overall mean lifetime of a drill tip is less than 32 feet Exercise 4 Hypothesis Test for p In a random sample of 160 students from Faber College 56 believe that resume inflation i e misrepresenting yourself on your resume is unethical a Find the p value of the test H0 p 0 40 versus H1 p 0 40 Solution Solution Solution Step 1 p the proportion of all Faber College students who believe that resume inflation is unethical H0 p 0 40 vs H1 p 0 40 Step 2 the sample proportion Normal p p 1 p n We will use cid 3404 cid 3043 cid 3548 cid 2879 cid 3043 cid 3493 cid 3043 cid 4666 cid 2869 cid 3043 cid 4667 cid 3041 Step 3 cid 3404 cid 2873 cid 2874 cid 2869 cid 2874 cid 2868 cid 3404 0 35 cid 3398 0 35 cid 3398 0 40 cid 3404 cid 3404 cid 3493 cid 4666 1 cid 4667 cid 3493 0 40 cid 4666 0 60 cid 4667 160 cid 3404 cid 3398 0 05 0 0387 cid 3404 cid 3398 1 29 p value P Z 1 29 0 985 b Is there significant evidence to suggest that fewer than 2 out of every 5 students think resume inflation is unethical at the 0 10 level of significance Step 4 Since the p value is less than 0 10 we reject the null hypothesis Step 5 Yes there is significant evidence to suggest that fewer than 2 out of every 5 students p 0 40 think resume inflation is unethical Exercise 5 Hypothesis Test for p An economist states that 10 of Champaign Urbana s labor force is unemployed A random sample of 400 people in the labor force is obtained of whom 28 are unemployed a Test whether Champaign Urbana s unemployment rate is as the economist claims or is different than the claim at a 1 level of significance using the critical region method for evidence Step 1 p the proportion of all Champaign Urbana residents who are unemployed H0 p 0 10 vs H1 p 0 10 Step 2 the sample proportion Normal p p 1 p n We will use cid 3404 cid 3043 cid 3548 cid 2879 cid 3043 cid 3493 cid 3043 cid 4666 cid 2869 cid 3043 cid 4667 cid 3041 Step 3 cid 3404 cid 2870 cid 2876 cid 2872 cid 2868 cid 2868 cid 3404 0 07 cid 3398 0 07 cid 3398 0 10 cid 3404 cid 3404 cid 3493 cid 4666 1 cid 4667 cid 3493 0 10 cid 4666 0 90 cid …

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# UIUC STAT 400 - Week 12 ‐ Discussion Questions

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