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Section 8 – 3C Testing a Claim About the Population Mean Examples Example 1 (One Tail Left Tail Test) A local radio station claims that it plays 55 minutes of music every hour. (More Music Less Noise). After several complaints the State Broadcasting Board conducts an investigation. The Board selects 41 random hours of the stations broadcasts and finds that the 41 broadcasts had an average of 53.4 minutes of music with a standard deviation of 3.8 minutes. Use a .01 significance level to test the claim that the average minutes of music broadcast each hour is less than 55 minutes. H 0: µ = 55 Sample Mean: x = 53.4 H1: µ < 55 (One Tail Test Left Tail) Sample Standard Deviation: sx = 3.8 n = 41 DF = 40 α = .01 (all .01 in the left tail) Graph of Critical information: Test Statistic: t= t= ( x − µ) sx n (53.4 − 55) 3.8 41 t = −2.70 Conclusion based on H 0 : Reject H0 Conclusion based on the problem: There is sufficient evidence at the α = .01 level to support the claim that the average minutes of music broadcast each hour is less than 55 minutes. t Distribution Degrees of Freedom 40 Area In One Tail (Right Tail) 0.100 0.050 0.025 0.010 0.005 1.303 1.684 2.021 2.423 2.704 If the Right Tail Critical Value is +2.423 then the the Left Tail Critical Value is –2.423 Section 8 – 3C Page 1 ©2013 Eitel Testing a Claim About the Population Mean Example 2 (One Tail Right Tail Test) Avis Car Rentals reports that the average daily rental cost for a midsize car at airports used for national flights is $ 47.56 per day base price without taxes, surcharges or other local costs. The Conde Nashʼs travel magazine feels that this price is too low. They donʼt want travelers to read about the low cost of car rentals if indeed the average cost is higher. A Conde Nash reporter checks the rates on mid sized cars at 35 airport rentals selected at random. The sample shows that the average cost is $ 51.23 a day with a standard deviation of $ 8.12. Test the claim at a .10 significance level that the average daily rental cost for a midsize car at airports used for national flights is more than $ 47.56 per day. H 0: µ = 47.56 Sample Mean: x = 51.23 H1: µ > 47.56 (One Tail Teat Right Tail) Sample Standard Deviation: sx = 8.12 n = 35 DF = 34 α = .10 (all .10 in the right tail) Graph of Critical information: Test Statistic: t= t= (x − µ) sx n (51.23− 47.56) 8.12 35 t = 2.67 Conclusion based on H 0 : Reject H0 Conclusion based on the problem: There is sufficient evidence at the α = .10 level to support the claim that the average daily rental cost for a midsize car at airports used for national flights is more than $ 47.56 per day t Distribution Degrees of Freedom 34 Section 8 – 3C Area In One Tail (Right Tail) 0.100 0.050 0.025 0.010 0.005 1.307 1.691 2.032 2.441 2.728 Page 2 ©2013 Eitel Testing a Claim About the Population Mean Example 3 (Two Tail Test) The IRS Audit division reports that the average money deducted for donations to charities in 2008 tax year was $ 245. The California Franchise Tax Board thinks that this amount is incorrect for the 2008 Californian returns. They select 16 California tax returns at random and find that the average donation returns was $ 240 with a standard deviation of $ 14. Test the claim at a .05 significance level that the average California tax return for charitable donations was not equal to $ 245. Assume that the distribution of charitable contributions in the 2008 state tax returns is normally distributed. H 0: µ = 245 Sample Mean: x = 240 H1: µ ≠ 245 (Two Tail Test) Sample Standard Deviation: sx = 14 n = 16 DF = 15 α = .05 ( α 2 = .025in each tail) Graph of Critical information: Test Statistic: t= t= (x − µ) sx n (240 − 245) 14 16 t = −1.43 Conclusion based on H 0 : Do not Reject H0 Conclusion based on the problem: There is not sufficient evidence at the α = .05 level to reject the hypothesis that the average 2008 California tax return for charitable donations was equal to $ 245 Distribution Degrees of Freedom 15 Section 8 – 3C Area In One Tail (Right Tail) 0.100 0.050 0.025 0.010 0.005 1.341 1.753 2.131 2.602 2.947 Page 3 ©2013 Eitel