i. Test Ratio: or for large samplesii. Critical Value: or for large samples (from table 3) .iii. Confidence Interval: or for large samples252y0552 10/31/05 (Open in ‘Print Layout’ format) ECO252 QBA2 Name _ Key______ FIRST HOUR EXAM Hour of class registered _____ October 17-18 2005 Class attended if different ____Show your work! Make Diagrams! Exam is normed on 50 points. Answers without reasons are not usually acceptable.I. (8 points) Do all the following. If you do not use the standard Normal table, explain! 6,5.4~ Nx Make diagrams. For x draw a Normal curve with a vertical line in the center at 4.5. For z draw a Normal curve with a vertical line in the center at zero.1. 35.17 xP 65.4365.45.17zP 25.067.3 zP 4012.0987.4999.025.0067.3 zPzP1252y0552 10/31/05 (Open in ‘Print Layout’ format)2. 00.75.4 xP 65.4765.45.4zP 1628.42.00 zP3. 22.10xP 65.422.10zP 95.0 zP 1711.3289.5.95.000 zPzP2252y0552 10/31/05 (Open in ‘Print Layout’ format)4. 06.x There is no excuse for not being able to do this at this point, since you should have done it 3 timesin the take-home.To find 06.z make a Normal diagram for z showing a mean at 0 and 50% above 0, divided into 6% above 06.z and 44% below 04.z. So 4400.006. zzPThe closest we can come is 4394.55.10 zP or 4406.56.10 zP. So use 555.106.z. 83.136555.15.406.06.zx3252y0552 10/31/05 (Open in ‘Print Layout’ format)II. (5 points-2 point penalty for not trying part a.) A dealer wishes to test the manufacture’s claim that the Toyota Caramba gets 28 miles per gallon. The data below is found. (Recomputing what I’ve done for you isa great way to waste time.)a. Compute the sample standard deviation,s, of the miles per gallon. Show your work! (2)b. Is the population mean significantly different (using a 95% confidence level) from 28 mpg? Show your work! (3)c. (Extra Credit) Find an approximate p value for your null hypothesis. (2)Row x 2x 1 23.56 555.074 2 26.09 680.688 3 27.55 759.003 4 26.92 724.686 5 29.20 852.640 6 29.02 842.160 7 22.48 505.350 8 27.45 753.503 9 25.90 670.810 10 25.81 666.156 sum 263.98 7010.070Solution: a. Compute the sample standard deviation,s, of the miles per gallon. Show yourwork! (2)398.261098.263nxx 9398.261007.701012222nxnxs61400.4952596.41148021.261400.4 s. Note that excessive rounding can throw this answer way off. Using 26x, I got778.2792509676070109)26(10701022s and 3541.5s.b. Is the population mean significantly different (using a 95% confidence level) from 28 mpg? Show your work! (3) Given: 398.26x ,14802.2s 7n and .05. 91 nDF and 262.29025.t.We are testing 28:0H. Use only one of the following!Confidence Interval: .6793.0461400.01061400.41014802.2nssx 5366.1398.266793.0262.2398.262xstx or 21.861 to 27.935. Since 28 is not on the confidence interval, it is significantly different from the sample mean.Critical Value: .6793.0461400.01061400.41014802.2nssx 5366.1286793.0262.2282xcvstx or 26.46 to 29.53. Since 26.398 is not between the critical values, it is significantly different from the population mean.Test Ratio: .6793.0461400.01061400.41014802.2nssx358.26793.028398.260xsxt. The ‘accept’ zone is between -2.262 and 2.262. Since -2.197 is not between these values, reject the null hypothesis.c. (Extra Credit) Find an approximate p value for your null hypothesis. (2)2.358 is between 262.29025.t and 821.2901.t. Since 358.22 tPpval, it is between .02 and .05.4252y0552 10/31/05 (Open in ‘Print Layout’ format)III. Do all of the following problems (2 each unless marked otherwise adding to 18+ points). Show your work except in multiple choice questions. (Actually – it doesn’t hurt there either.) If the answer is ‘None of the above,’ put in the correct answer.[10]1. For sample sizes greater than 100, the sampling distribution of the mean will be approximately normally distributeda) *Regardless of the shape of the population.b) Only if the shape of the population is symmetrical.c) Only if the standard deviation of the samples are known.d) Only if the population is normally distributed.2. When determining the sample size for a proportion for a given level of confidence and sampling error, the closer to 0.50 that p is estimated to be the __________ the sample size required.a) Smallerb) *Largerc) Sample size is not affected.d) The effect cannot be determined from the information given.3. Which of the following would be an appropriate null hypothesis?a) The population proportion is less than 0.65. b) The sample proportion is less than 0.65.c) *The population proportion is at most 0.65.d) The sample proportion is at most 0.65.4. A Type I error is committed whena) *We reject a null hypothesis that is true.b) We don't reject a null hypothesis that is true. Note that b) and c) are not errors.c) We reject a null hypothesis that is false.d) We don't reject a null hypothesis that is false5. If we are performing a two-tailed test of whether = 100, the power of the test in detecting a shift of the mean to 105 will be ________ its power detecting a shift of the mean to 96.a) *Less than 96 is closer than 105 to 100.b) Greater thanc) Equal tod) Not comparable to6. From a sample of 100 students we find that the mean expenditure for books is $316.40 with a sample standard deviation of $43.20. You are asked to test whether the (population) mean expenditure is above $314 using a 10% significance level.a) What are the null and alternative hypotheses? (2)b) What is the ‘rejection zone’ (in terms of x) (2)c) What is your conclusion? (2) [16]Solution: This is basically the revised version of Exercise 9.54 [9.52 in 9th] (9.50 in 8th edition) that I warned you about. a) Since the problem asks if the mean is above $314 314, and this does not contain an equality,it must be an alternate hypothesis. Our
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