Workshop 9 Solutions(2) - Workshop 9 Solutions THE...

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Workshop 9 Solutions T HE U NIVERSITY OF S YDNEY Q UANTITATIVE B USINESS A NALYSIS Prepared by: Scott Liu Workshop Aims 1. Understand the type of hypotheses that we test and how to select the null and alternative hypotheses. 2. Understand the difference between a one-tail test and a two-tail test. 3. Understand the types of errors we can make when we test hypotheses. 4. Use the 6 step procedure to undertake a hypothesis test for a mean. 5. Write a one sentence report (other than reject H 0 ) relating the conclusion back to the problem. 6. Calculate the p -value for a test statistic.
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BUSS1020 Quantitative Business Analysis Workshop 9 Solutions Workshop 9 Question 1 For each of the following tests of hypotheses about μ , determine the rejection region if σ is known. (a) H 0 : μ = 100, H 1 : μ 6= 100, α = 0.05 (b) H 0 : μ = 50, H 1 : μ > 100, α = 0.01 (c) H 0 : μ = 15, H 1 : μ < 100, α = 0.1 Solution (a) This is a two-tailed hypothesis test with a 5% level of significance. Using NORM . INV ( 0 . 025 , 0 , 1 ): Reject if: | Z stat | > 1.96 (b) This is an upper-tail hypothesis test with a 1% level of significance. Using NORM . INV ( 0 . 01 , 0 , 1 ): Reject if: Z stat > 2.326 (c) This is a lower-tail hypothesis test with a 10% level of significance. Using NORM . INV ( 0 . 1 , 0 , 1 ): Reject if: Z stat < - 1.282 Note | Z stat | > 1.96 is the same as saying Z stat > 1.96 and Z stat < - 1.96. Question 2 For each of the following parts, determine the test statistic and rejection region. (a) H 0 : μ = 100, H 1 : μ 6= 100, α = 0.05, ¯ x = 90, s = 20, n = 40 (b) H 0 : μ = 50, H 1 : μ > 50, α = 0.01, ¯ x = 53, s = 10, n = 80 (c) H 0 : μ = 15, H 1 : μ < 15, α = 0.1, ¯ x = 13.5, s = 20, n = 240 Solution (a) This is a two-tailed hypothesis test with a 5% level of significance and degree of freedom 40 - 1 = 39. Using T . INV ( 0 . 025 , 39 ): Reject if: | t stat | > 2.023 (b) This is an upper-tail hypothesis test with a 1% level of significance and degree of freedom 80 - 1 = 79. Using T . INV ( 0 . 01 , 79 ): Reject if: t stat > 2.374 (c) This is a lower-tail hypothesis test with a 10% level of significance and degree of freedom 240 - 1 = 239. Using T . INV ( 0 . 1 , 239 ): Reject if: t stat < - 1.285 University of Sydney Page 2
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BUSS1020 Quantitative Business Analysis Workshop 9 Solutions Question 3 For each of the following parts, state the null and alternative hypotheses. (a) The manager of the Danvers-Hinton Resort believes that the mean guest bill is more than $500.00. (b) A long distance trucking firm believes that its mean weekly loss due to damaged shipments is less than $2000. (c) An automobile assembly line operation has a scheduled mean completion time of 2.2 minutes. (d) Historically, long distance telephone calls from Sydney have averaged 15.20 minutes per call. (e) Gross Domestic Product has grown by less than 4% per year over the last 30 years. (f) The mean weekly rental fee for second-floor two-bedroom apartments in the city is more than $400. (g) The mean family size in New South Wales is less than 4.3. Solution (a) H 0 : μ = 500, H 1 : μ > 500 (b) H 0 : μ = 2000, H 1 : μ < 2000 (c) H 0 : μ = 2.2, H 1 : μ 6= 2.2 (d) H 0 : μ = 15.20, H 1 : μ 6= 15.20 (e) H 0 : μ = 0.04, H 1 : μ < 0.04 (f) H 0 : μ = 400, H 1 : μ > 400 (g) H 0 : μ = 4.3, H 1 : μ < 4.3 Solution (1) The null hypothesis is always have the ‘=’ sign.
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