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 x N 4 11 0 4 P z 0 36 0 36 z 0 P z 0 1406 5 6406 1 P x 0 P z 11 2 4 33 4 z P 3 36 z 0 18 P 3 36 z 0 2 P 33 x 2 P 11 11 P 0 18 z 0 4996 0714 4282 1 252y0753 10 19 07 Open in Print Layout format 4 4 4 4 z P 0 73 z 0 2673 3 P 4 x 4 P 11 11 4 x 075 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 z 075 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 z 075 and zero is 92 5 50 42 5 or P 0 z z 075 4250 If we check this against the Normal table the closest we can come to 4250 is P 0 z 1 44 4251 So z 075 1 44 This is the value of z that you need for a 85 confidence interval To get from z 075 to x 075 use the formula x z which is the opposite of z x x 4 1 44 11 19 84 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 x 075 it must be above the mean and have 42 5 of the distribution between it and the mean 19 84 4 Check P x 19 84 P z P z 1 44 P z 0 P 0 z 1 44 11 5 4251 0749 075 This is identical to the way you normally get a p value for a rightsided test 2 252y0753 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 0 a 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 the mean 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 below 2 3 Solution a Compute the sample standard deviation s of expenditures The first two columns are needed for the Row x x x x2 computational shortcut method The first third 2 x x and fourth are needed for the definitional 1 2 5 6 25 0 98333 0 9669 method Using both methods or the 2 7 3 53 29 3 81667 14 5669 definitional method wastes time 3 3 1 9 61 0 38333 0 1469 4 2 6 6 76 0 88333 0 7803 x 20 9 x 2 90 67 5 6 2 4 3 0 20 9 x 5 76 9 00 90 67 1 08333 0 48333 0 0 1 1736 0 2336 17 8682 x 20 9 3 4833 s n 6 2 x x 2 x and nx 2 n 1 x 0 a check n 6 x x 2 544 90 67 6 3 4833 2 17 8697 3 5739 5 5 s x 3 5739 1 8905 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 x x 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 n 1 s 3 4833 4 032 0 77178 3 483 3 112 or 0 371 to 6 595 x 2 2 x t sx 1 8905 3 5739 5 t n 1 t 005 4 032 0 59565 0 77178 2 6 n 6 c Redo b when you find out that there were only 50 people living in Rough Corners 2 sx x t n 1 s x 136 00 4 604 4 3304 136 00 19 98 or 116 02 to 155 98 2 sx sx n 1 8905 N n N 1 6 50 6 50 1 3 5739 44 9 0 59565 0 41237 0 6422 6 49 13 d Assume that the population standard deviation is 2 and create an 85 two sided confidence interval for the mean 2 2 We found z 075 1 44 on the last page We have 2 n 6 x 3 4833 and x x n 2 6 4 0 66667 0 8165 6 x z 3 x1 3 4833 1 44 0 8165 3 4833 1 1758 or 2 3075 to 4 6591 3 252y0753 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 x 3 4833 n 6 and s x sx n 0 77178 Since this is a one sided hypothesis we will use 5 t n 1 t 01 3 365 Needless to say because of the small sample size we are assuming that the parent distribution is Normal Our hypotheses are H 0 2 3 so H 1 2 3 0 2 3 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 t x 0 3 4833 2 3 1 5332 This is a right sided test the larger the sample sx 0 77178 mean is the more positive will be this ratio We will reject the null hypothesis if the ratio is larger than 5 t n 1 t 01 3 365 Make a 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