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1、Hypothesis TestingChapter 9Hypothesis TestingLearning Objectives1.Learn how to formulate and test hypotheses about a population mean and/or a population proportion.2.Understand the types of errors possible when conducting a hypothesis test.3.Be able to determine the probability of making various err
2、ors in hypothesis tests.4.Know how to compute and interpret p-values.5.Be able to determine the size of a simple random sample necessary to keep the probability of hypothesis testing errors within acceptable limits.6.Know the definition of the following terms:null hypothesisone-tailed testalternativ
3、e hypothesistwo-tailed testtype I errorp-valuetype II erroroperating characteristic curvecritical valuepower curvelevel of significanceSolutions:1.a.H0:m 600Managers claim.Ha:m 600b.We are not able to conclude that the managers claim is wrong.c.The managers claim can be rejected. We can conclude tha
4、t m 400.2.a.H0:m 14Ha:m 14Research hypothesisb.There is no statistical evidence that the new bonus plan increases sales volume.c.The research hypothesis that m 14 is supported. We can conclude that the new bonus plan increases the mean sales volume.3.a.H0:m = 32Specified filling weightHa:m 32Overfil
5、ling or underfilling existsb.There is no evidence that the production line is not operating properly. Allow the production process to continue.c.Conclude m 32 and that overfilling or underfilling exists. Shut down and adjust the production line.4.a.H0:m 220Ha:m 220Research hypothesis to see if mean
6、cost is less than $220.b.We are unable to conclude that the new method reduces costs.c.Conclude m 1b.Claiming m 1 when it is not. This is the error of rejecting the products claim when the claim is true.c.Concluding m 1 when it is not. In this case, we miss the fact that the product is not meeting i
7、ts label specification.7.a.H0:m 8000Ha:m 8000Research hypothesis to see if the plan increases average sales.b.Claiming m 8000 when the plan does not increase sales. A mistake could be implementing the plan when it does not help.c.Concluding m 8000 when the plan really would increase sales. This coul
8、d lead to not implementing a plan that would increase sales.8.a.H0:m 220Ha:m 220b.Claiming m 220 when the new method does not lower costs. A mistake could be implementing the method when it does not help.c.Concluding m 220 when the method really would lower costs. This could lead to not implementing
9、 a method that would lower costs.9.a.z = -1.645Reject H0 if z 2.05b.c.Area from z = 0 to z = 1.36 is .4131p-value = .5000 - .4131 = .0869d.Do not reject H011.Reject H0 if z -1.645a. Reject H0b. Do not reject H0 c. Do not reject H0d. Reject H012.a.p-value = .5000 - .4656 = .0344Reject H0b.p-value = .
10、5000 - .1736 = .3264Do not reject H0c.p-value = .5000 - .4332 = .0668Do not reject H0d.z = 3.09 is the largest table value with .5000 - .4990 = .001 area in tail. For z = 3.30, the p-value is less than .001 or approximately 0. Reject H0.e.Since z is to the left of the mean and the rejection region i
11、s in the upper tail, p-value = .5000 + .3413 = .8413. Do not reject H0.13.a.H0:m 1056Ha:m 1056 b.Reject H0 if z 2.33b.c.Reject H0; conclude the mean television viewing time per day is greater than 6.70.15.a.z.05 = 1.645Reject H0 if z -1.645b.c.Reject H0; conclude that the mean sales price of used ca
12、rs is less than the national average.16.a.H0:m 13Ha:m 13b.z.01 = 2.33Reject H0 if z 15b.c.p-value = .5000 - .4985 = .0015d.Reject H0; the premium rate should be charged.18.a.H0:m 5.72Ha:m 5.72b.c. p-value = .5000 - .4830 = .0170d.p-value a; reject H0. Conclude teens in Chicago have a mean expenditur
13、e greater than 5.72. 19.a.H0:m 181,900Ha:m 181,900b.c.p-value = .5000 - .4983 = .0017d.p-value 37,000b.c. p-value = .5000 - .4292 = .0708d.p-value a; do not reject H0. Cannot conclude population mean salary has increased in June 2001.21.a.Reject H0 if z 1.96b.Reject H0; conclude Ha is true.22.a.Reje
14、ct H0 if z 2.33b.c.p-value = (2) (.5000 - .3708) = .2584d.Do not reject H023.Reject H0 if z 1.96a. Reject H0b. Do not reject H0c. Do not reject H0d. Reject H024.a.p-value = 2 (.5000 - .4641) = .0718Do not reject H0b.p-value = 2(.5000 - .1736) = .6528Do not reject H0c.p-value = 2(.5000 - .4798) = .04
15、04Reject H0d.approximately 0Reject H0e.p-value = 2(.5000 - .3413) = .3174Do not reject H025. a.z.025 = 1.96Reject H0 if z 1.96b.c.Do not reject H0. Cannot conclude a change in the population mean has occurred.d.p-value = 2(.5000 - .4382) = .123626.a.H0:m = 8Ha:m 8Reject H0 if z 1.96b.c.Do not reject
16、 H0; cannot conclude the mean waiting time differs from eight minutes.27.a.H0:m = 16Continue productionHa:m 16Shut downReject H0 if z 1.96b.Reject H0 and shut down for adjustment.c.Do not reject H0; continue to run.d.For = 16.32, p-value = 2 (.5000 - .4857) = .0286For = 15.82, p-value = 2 (.5000 - .
17、3907) = .218628.a.H0:m = 1075Ha:m 1075b.c.p-value = 2(.5000 - .4236) = .1528d.Do not reject H0. Cannot conclude a change in mean amount of charitable giving.29.a.H0:m = 15.20Ha:m 15.20b.p-value = 2(.5000 - .3554) = .2892c.Do not reject H0; the sample does not provide evidence to conclude that there
18、has been a change.30.a.H0:m = 26,133Ha:m 26,133b.c.p-value = 2(.5000 - .4817) = .0366d.p-value a; reject H0. Conclude population mean wage in Collier County differs from the state mean wage.31.a.935 25 or 910 to 960Since 900 is not in the interval, reject H0 and conclude m 900.b.Reject H0 if z 1.96R
19、eject H0c.p-value = 2(.5000 - .4970) = .006032.a.The upper 95% confidence limit is computed as follows:Thus, we are 95% confident that m is $14.66 per hour or less.b.Since $15.00 is not in the interval $14.66 per hour or less, we reject H0.Conclude that the mean wage rate is less than $15.00.33.a.Wi
20、th 15 degrees of freedom, t.05 = 1.753Reject H0 if t 1.753b.Do not reject H034.a. = 108 / 6 = 18b.c.Reject H0 if t 2.571d.e.Reject H0; conclude Ha is true.35. Reject H0 if t -1.721a.Do not reject H0 b.Reject H0c.Do not reject H0d.Do not reject H036.Use the t distribution with 15 degrees of freedoma.
21、p-value = .01Reject H0b.p-value = .10Do not reject H0c.p-value is between .025 and .05Reject H0d.p-value is greater than .10Do not reject H0e.p-value is approximately 0Reject H037.a.H0:m = 3.00Ha:m 3.00b.t.025 = 2.262Reject H0 if t 2.262c.d.e.f.Do not reject H0; cannot conclude the population means
22、earning per share has changed.g.t.10 = 1.383p-value is greater than .10 x 2 = .20Actual p-value = .391638.a.t.025 = 2.064 24 degrees of freedomReject H0 if t 2.064b.c.Do not reject H0; cannot conclude the mean expenditure in Corning differs from the U.S. mean expenditure.39.a.t.05 = 1.895 7 degrees
23、of freedomb.c.d.c.Reject H0; conclude that the mean number of hours worked exceeds 55.40.a.H0:m = 4000Ha:m 4000b.t.05 = 2.160 13 degrees of freedomReject H0 if t 2.160c.d.Do not reject H0; Cannot conclude that the mean cost in New City differs from $4000.e.With 13 degrees of freedomt.05 = 1.771t.10
24、= 1.3501.63 is between 1.350 and 1.771. Therefore the p-value is between .10 and .20.41. a.H0:m 280Ha:m 280b.286.9 - 280 = 6.9 yardsc.t.05 = 1.860 with 8 degrees of freedomd.e.Reject H0; The population mean distance of the new driver is greater than the USGA approved driver.f.t.05 = 1.860t.025 = 2.3
25、06p-value is between .025 and .05Actual p-value = .036142.a.H0:m 2Ha:m 2b.With 9 degrees of freedom, reject H0 if t 1.833c. = 24 / 10 = 2.4d.e.f.Reject H0 and claim m is greater than 2 hours. For cost estimating purposes, consider using more than 2 hours of labor time.g.t.025 = 2.262, t.01 = 2.821p-
26、value is between .025 and .01.43.a.Reject H0 if z 1.645b.Reject H044.a.Reject H0 if z 1.96b.c.p-value = 2 (.5000 - .3944) = .2122d.Do not reject H0.45.Reject H0 if z .40b. Reject H0 if z 1.645c.= 188/420 = .4476d.Reject H0. Conclude that there has been an increase in the proportion of users receivin
27、g more than ten e-mails per day. 47.a.z.05 = 1.645Reject H0 if z -1.645b.= 52/100 = .52c.Reject H0. Conclude less than 64% of shoppers believe supermarket ketchup is as good as the national brand.d.p-value = .5000 - .4938 = .006248.a.= 285/500 = .57b.c.z = 3.13 is not in the table. Closest value is
28、z = 3.09. Thus, p-value is approximately.5000 - .4990 = .001d.p-value .01, Reject H0. Over 50% prefer Burger King fries.e.Yes; the statistical evidence shows Burger King fries are preferred. The give-away was a good way to get potential customers to try the new fries. 49.a.H0:p = .48Ha:p .48b. z.025
29、 = 1.96Reject H0 if z 1.96c.= 368/800 = .45d.d.Do not reject H0. Cannot conclude the proportion of drivers who do not stop has changed. 50.a.= 67/105 = .6381 (about 64%)b.c.p-value = 2(.5000 - .4977) = .0046d.p-value .15Reject H0 if z 2.33= 88/500 = .176Do not reject H0; p .15 cannot be rejected. Th
30、us the special offer should not be initiated.p-value = .5000 - .4484 = .051654.a.H0:p .047Ha:p .047b.= 35/1182 = .0296c.d.p-value = .5000 - .4976 = .0024e.p-value a, reject H0. The error rate for Brooks Robinson is less than the overall error rate. 55.H0:p .20Ha:p .20Reject H0 if z -1.645= 83/596 =
31、.1393p-value 0Reject H0; conclude that less than 20% of workers would work for less pay in order to have more personal and leisure time.56.c = 10 - 1.645 (5 /) = 9.25Reject H0 if 9.25a.When m = 9,Prob (H0) = (.5000 - .2088) = .2912b.Type II errorc.When m = 8,b = (.5000 - .4969) = .003157.Reject H0 i
32、f z 1.96c1 = 20 - 1.96 (10 /) = 18.61c2 = 20 + 1.96 (10 /) = 21.39a.m = 18b = .5000 - .3051 = .1949b.m = 22.5b = .5000 - .4418 = .0582c.m = 21b = .5000 +.2088 = .708858.a.H0:m 15Ha:m 15Concluding m 15 when this is not true. Fowle would not charge the premium rate even though the rate should be charged.b.Reject H0 if z 2.33Solve for = 16.58Decision Rule:Accept H0 if 16.58Reject H0 if 16.58For m = 17,b = .5000 -.2324 = .2676c.For m = 18, b = .5000 -.4821 = .017959.a.H0:m 25Ha:m 25Reject H0 if z -2.05Solve for = 23.88