Alternate formulas for this section include those below.252y0753 10/19/07 (Open in ‘Print Layout’ format) ECO252 QBA2 Name:_KEY___________ FIRST EXAM Class hour: _____________ October 4 and 8, 2007 Version 3 Student number: __________Show your work! Make Diagrams! Include a vertical line in the middle! Exam is normed on 50 points. Answers without reasons are not usually acceptable.I. (8 points) Do all the following. 11,4~ Nx1. 0xP 36.01140 zPzP 0036.0 zPz = .1406 + .5 = .64062. 233 xP 18.036.3114211433 zPzP 036.3 zP 018.0 zP=.4996 - .0714 = .42821252y0753 10/19/07 (Open in ‘Print Layout’ format)3. 44 xP 073.011441144 zPzP= .26734. 075.x (Do not try to use the t table to get this.)For z make a diagram. Draw a Normal curve with a mean at 0. 075.z is the value of z with 7.5% of the distribution above it. Since 100 – 7.5 = 92.5, it is also the .925 fractile. Since 50% of the standardized Normal distribution is below zero, your diagram should show that the probability between 075.z and zerois 92.5% - 50% = 42.5% or .4250.0075. zzP If we check this against the Normal table, the closest we can come to .4250 is .4251.44.10 zP So .44.1075.z This is the value of z that you need for a 85% confidence interval. To get from 075.z to 075.x, use the formulazx , which is the opposite of xz . 84.191144.14 x. If you wish, make a completely separate diagram for x. Draw a Normal curve with a mean at 4. Show that 50% of the distribution is below the mean (4). If 7.5% of the distribution is above 075.x, it must be above the mean and have 42.5% of the distribution between it and the mean.Check: 84.19xP 44.111484.19 zPzP 44.100 zPzP.075.0749.4251.5. This is identical to the way you normally get a p-value for a right-sided test.2252y0753 10/19/07 (Open in ‘Print Layout’ format)II. (9 points-2 point penalty for not trying part a.)Monthly incomes (in thousands) of 6 randomly picked individuals in the little town of Rough Corners are shown below. 2.5 7.3 3.1 2.6 2.4 3.0a. Compute the sample standard deviation,s, of expenditures. Show your work! (2)b. Assuming that the underlying distribution is Normal, compute a 99% confidence interval for themean. (2)c. Redo b) when you find out that there were only 50 people living in Rough Corners. (2)d. Assume that the population standard deviation is 2 and create an 85% two-sided confidence interval for the mean. (2)e. Use your results in a) to test the hypothesis that the mean income is above 2.3(thousand) at the 99% level. (3) State your hypotheses clearly!f. (Extra Credit) Given the data, test the hypothesis that the population standard deviation is below2. (3) Solution: a) Compute the sample standard deviation,s, of expenditures.Row x 2x xx 2xx 1 2.5 6.25 -0.98333 0.9669 2 7.3 53.29 3.81667 14.5669 3 3.1 9.61 -0.38333 0.1469 4 2.6 6.76 -0.88333 0.7803 5 2.4 5.76 -1.08333 1.1736 6 3.0 9.00 -0.48333 0.2336 20.9 90.67 0.0 17.8682 The first two columns are needed for the computational (shortcut) method. The first, third and fourth are needed for the definitional method. Using (both methods or) the definitional method wastes time.9.20x, 67.902x, 0xx(a check), 5442xx and 6n.4833.369.20nxx 5739.358697.1754833.3667.9012222nxnxsx8905.15739.3 xs. If you used the definitional method, you would have gotten 1.8904. There seems to be a lot of potential for rounding error here. Note that the xx column, even though it carries an extra place, does not quite add to the expected zero but to .00002.b) Assuming that the underlying distribution is Normal, compute a 99% confidence interval for the mean. (2) xnstx12 112.3483.377178.0032.44833.3 or 0.371 to 6.595.nssxx77178.059565.065739.368905.1 032.45005.12ttnc) Redo b) when you find out that there were only 50 people living in Rough Corners. (2) xnstx12 98.1900.1363304.4604.400.136 or 116.02 to 155.981NnNnssxx 15065068905.16422.041237.013959565.0494465739.3d) Assume that the population standard deviation is 2 and create an 85% two-sided confidence interval for the mean. (2) (2) We found 44.1075.z on the last page. We have 2, 6n, 4833.3x and8165.066667.06462nxx 1758.14833.38165.044.14833.313xzx or 2.3075 to 4.6591.3252y0753 10/19/07 (Open in ‘Print Layout’ format)e) Use your results in a) to test the hypothesis that the mean income is above 2.3(thousand) at the 99% level. (3) State your hypotheses clearly! The statement that the mean is above 2.3 does not contain an equality, so it must be an alternate hypothesis. We have the following information. 01.,4833.3x, 6n and nssxx77178.0. Since this is a one-sided hypothesis we will use 365.3501.1ttn. Needless to say, because of the small sample size, we are assuming that the parent distribution is Normal. Our hypotheses are 3.2:3.2:10HHso 3.20. Since we are worrying about the mean being too large, this is a right-sided test. There are three ways to do this. Do only one of them.(i) Test Ratio: 5332.177178.03.24833.30xsxt . This is a right-sided test - the larger the sample mean is, the more positive will be this ratio. We will reject the null hypothesis if the ratio is larger than 365.3501.1ttn. Make a diagram showing a Normal curve with a mean at 0 and a shaded 'reject' zone above 3.365. Since the test ratio is below 3.365, we cannot reject 0H.If you wish to find a p-value for your hypothesis, note that the t-ratio is 1.5332. The p-value will be the probability that t is above 3.365. The line of the t table for 5 degrees of freedom is below.
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