STAT 400 Lecture AL1 0 Examples for 8 1 Part 1 Spring 2015 Dalpiaz A car manufacturer claims that when driven at a speed of 50 miles per hour on a highway the mileage of a certain model follows a normal distribution with mean 0 30 miles per gallon and standard deviation 4 miles per gallon A consumer advocate thinks that the manufacturer is overestimating average mileage The advocate decides to test the null hypothesis H 0 30 against the alternative hypothesis H 1 30 a Suppose the actual overall average mileage is indeed 30 miles per gallon What is the probability that the sample mean is 29 4 miles per gallon or less for a random sample of n 25 cars 29 4 30 P X 29 4 P Z P Z 0 75 0 75 0 2266 4 25 b A random sample of 25 cars yields x 29 4 miles per gallon Based on the answer for part a is there a reason to believe that the actual overall average mileage is not 30 miles per gallon If 30 it is not unusual to see the values of the sample mean x at 29 4 miles per gallon or even lower It does not imply that 30 but we have no reason to doubt the manufacturer s claim c Suppose the actual overall average mileage is indeed 30 miles per gallon What is the probability that the sample mean is 28 miles per gallon or less for a random sample of n 25 cars 28 30 P X 28 P Z P Z 2 50 2 50 0 0062 4 25 d A random sample of 25 cars yields x 28 miles per gallon Based on the answer for part c is there a reason to believe that the actual overall average mileage is not 30 miles per gallon If 30 it is very unusual to see the values of the sample mean x at 28 miles per gallon or lower It does not imply that 30 but we have a very good reason to doubt the manufacturer s claim e Suppose the consumer advocate tests a sample of n 25 cars What is the significance level associated with the rejection region Reject H 0 if x 28 6 significance level P Type I Error P Reject H 0 H 0 true X Need P X 28 6 30 Z n 28 6 30 P X 28 6 30 P Z P Z 1 75 1 75 0 0401 4 25 f Suppose the consumer advocate tests a sample of n 25 cars Find the rejection region with the significance level 0 05 0 05 n 25 Rejection Region Reject H 0 if Z X 0 z Z n X 30 1 645 4 X 30 1 645 4 25 28 684 25 g Suppose that the sample mean is x 29 miles per gallon for a sample of n 25 cars Find the p value of the appropriate test H 0 30 Z X 0 n vs H 1 30 29 30 1 25 4 25 P Z 1 25 1 25 0 1056 Left tailed h State your decision Accept H 0 or Reject H 0 for the significance level 0 05 P value Since 0 1056 0 05 i P value Do NOTReject H 0 Reject H 0 Do NOT Reject H 0 at 0 05 Construct a 95 confidence interval for the overall average miles per gallon rating for this model 4 is known n 25 The confidence interval X z 2 95 confidence level 29 1 96 j 4 0 05 2 0 025 29 1 568 25 n z 1 96 2 27 432 30 568 What is the minimum sample size required if we want to estimate to within 0 5 miles per gallon with 95 confidence 0 5 4 95 confidence level 0 05 z 2 n k 2 2 0 025 1 96 4 245 8624 0 5 The confidence upper bound for X z 29 1 645 4 25 2 2 Round up Construct a 95 confidence upper bound for 95 confidence level z 1 96 n 0 05 z 1 645 29 1 316 0 30 316 n 246 1 The overall standard deviation of the diameters of the 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 test using a 10 level of significance Claim 4 300 H 0 4 300 Test Statistic Z X 0 H 1 4 300 vs n is known 4 299 4 300 1 80 0 005 81 2 tailed Rejection Region Reject H 0 if Z z 0 10 2 or Z z 2 z 0 05 1 645 2 0 05 Reject H 0 if Z 1 645 or Z 1 645 Decision The value of the test statistic does fall into the Rejection Region Reject H 0 at 0 10 OR P value p value P Z 1 80 P Z 1 80 0 0359 0 0359 0 0718 Decision 0 0718 0 10 P value Reject H 0 at 0 10 OR Confidence Interval is known The confidence interval X z 4 299 1 645 0 10 90 conf level 0 005 2 0 05 2 z 0 05 1 645 4 299 0 0009138889 81 Decision 90 confidence interval for does not cover 4 300 Reject H 0 at 0 10 b Two tailed test same Confidence Interval Accept H 0 Covers 0 Reject H 0 Does not cover 0 State your decision Accept H 0 or Reject H 0 for the significance level 0 05 0 0718 0 05 P value Do NOT Reject H 0 Accept H 0 at 0 05 n 2 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 a Perform the appropriate test Use the significance level 0 10 1800 Claim H 0 1800 Test Statistic T H 1 1800 vs X 0 s n is unknown 1921 54 1800 2 485 249 39 26 Right tailed Rejection Region Reject H 0 if T t 0 10 n 1 25 degrees of freedom t 0 10 1 316 Reject H 0 if T 1 316 Decision The value of the test statistic does fall into the Rejection Region Reject H 0 at 0 10 OR P value P value Area to the right of T 2 485 0 01 n 1 25 degrees of freedom Decision 0 01 0 10 P value Reject H 0 at 0 10 b State your decision Accept H 0 or Reject H 0 for the significance level 0 05 P value 0 01 0 05 Reject H 0 at 0 05 …
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