i. Test Ratio: or for large samplesii. Critical Value: or for large samples (from table 3) .iii. Confidence Interval: or for large samples252y0551 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! 6,5.2~ Nx Make diagrams. For x draw a Normal curve with a vertical line in the center at 2.5. For z draw a Normal curve with a vertical line in the center at zero.1. 35.19 xP 65.2365.25.19zP 08.067.3 zP 5318.0319.4999.08.00067.3 zPzP1252y0551 10/31/05 (Open in ‘Print Layout’ format)2. 50.20 xP 65.25.265.20zP 042.0 zP1628.3. 22.10xP 65.222.10zP 29.1 zP 0985.4015.5.29.100 zPzP2252y0551 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. 83.116555.15.206.06.zxFor the method of creating the diagrams, go to 252y0551s3252y0551 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 28.82 830.592 2 23.71 562.164 3 29.25 855.563 4 25.16 633.026 5 27.60 761.760 6 27.41 751.308 7 27.23 741.473 8 26.42 698.016 9 21.75 473.063 10 26.67 711.289sum 264.02 7018.253 Solution: a. Compute the sample standard deviation,s, of the miles per gallon. Show your work!(2)402.261002.264nxx 9402.2610253.701812222nxnxs28855.5959696.47299685.228855.5 s. Note that excessive rounding can throw this answer way off. Using 26x, I got667.2892589676070189)26(10701822s and 3541.5s.b. Is the population mean significantly different (using a 95% confidence level) from 28 mpg? Show your work! (3)Given: 402.26x ,2997.2s 7n and .05. 91 nDF and 262.29025.t.We are testing 28:0H. Use only one of the following!Confidence Interval: .7272.052855.01028855.5102997.2nssx 6449.1402.267272.0262.2402.262xstx or 24.757 to 28.047. Since 28 is on the confidence interval, it is not significantly different from the sample mean.Critical Value: .7272.052855.01028855.5102997.2nssx 6449.1287272.0262.2282xcvstx or 26.36 to 29.65. Since 26.402 is between the critical values, it is not significantly different from the population mean.Test Ratio: .7272.052855.01028855.5102997.2nssx197.27272.028402.260xsxt. The ‘accept’ zone is between -2.262 and 2.262. Since -2.197 is between these values, do not reject the null hypothesis.c. (Extra Credit) Find an approximate p value for your null hypothesis. (2)2.197 is between 262.29025.t and 833.1905.t. Since 197.22 tPpval, it is between .05 and .10.4252y0551 10/31/05 (Open in ‘Print Layout’ format)For Minitab computation of variance and hypothesis test, go to 252y0551s5252y0551 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.1. For sample sizes less than 10, 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 sample is normally distributed.2. The width of a confidence interval estimate for a proportion will bea) Narrower for 99% confidence than for 95% confidence.b) Wider for a sample size of 100 than for a sample size of 50.c) *Narrower for 90% confidence than for 95% confidence.d) Narrower when the sample proportion is 0.50 than when the sample proportion is 0.203. Which of the following would be an appropriate null hypothesis?a) *The mean of a population is equal to 55. b) The mean of a sample is equal to 55.1. The mean of a population is greater than 55.2. Only (a) and (c) are true.4. A Type II 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 false.5. 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 94.a) *Less than 94 is further from 100 than 105.b) Greater thanc) Equal to3. Not comparable to [10]6. From a sample of 100 students we find that the mean expenditure for books is $316.40 with a standard deviation of $43.20. You are asked to test whether the (population) mean expenditure is above $302 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)Solution: This is basically the revised version of Exercise 9.54 [9.52 in 9th] (9.50 in 8th edition) that I warned
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