DIAGRAM2/22/99 252y9912 ECO252 QBA2 Name FIRST HOUR EXAM Hour of Class Registered (Circle) February 23, 1999 MWF TR 10 12 12:30 2:00Show your work! Make Diagrams!I. (14 points) Do all the following. 7,6~ Nx1. 31 xP 43.000.1763761 zPzP 1749.1664.3413.043.0000.1 zPzP2. 1111 xP 71.043.276117611 zPzP 7536.2611.4925.71.00043.2 zPzP3. 916 xP 43.014.37697616 zPzP 6656.1664.4992.43.00014.3 zPzPFor 3.14 see SS Table 18.4. 0xP 0086.086.0760 zPzPzPzP8051.5.3051. 5. 5.129 xP 93.043.0765.12769 zPzP 1574.1664.3238.43.0093.00 zPzP6. A symmetrical interval about the mean with 96% probability.We want two points 02.98. and xx, so that 9600.02.98. xxxP. From the diagram, if we replace x by z, 4800.002. zzP. The closest we can come is 4798.05.20 zPor 4803.06.20 zP. So 054.202.z, though 2.05 or 2.06 are fine, and 38.1467054.2602.zx, or –8.38 to 20.38.7. 13.xWe want a point 13.x, so that 13.13.xxP. From the diagram, if we replace x by z, 3700.013. zzP. The closest we can come is 3708.13.10 zP. So 13.113.z, and 91.76713.1611.zx, or 13.91 .2/22/99 252y9912II. (6 points-2 point penalty for not trying part a.) A sample of the number of days it took a broker to sell houses appears below. Assume that we were sampling from a normal distribution. Home Days 1 49 2 69 3 88 4 99 5 33 a. Compute the sample variance, s, of the number of days it takes to sell a house. Show your work.(3)b. Compute an 80% confidence interval for the mean time, , that it takes to sell a house.(3)Solution: a. Home x (Days) x2 1 49 2401 2 69 4761 3 88 7744 4 99 9801 5 33 1089 Total 338 25796. b. From the problem statement 20.. From Table 3 of the syllabus supplement, if the population variance is unknown x t sx2 and 533.1410.12ttn.1392.1251441.2758.736nssx. So 61.186.671372.12533.16.67 or 48.99 to 86.21.2 6.675338nxx 46.6752579612222nxnxs 8.736 or 1441.27s.2/22/99 252y9912III. Do at least 3 of the following 5 Problems (at least 10 each) (or do sections adding to at least 30 points - Anything extra you do helps, and grades wrap around) . Show your work! State 0H and 1H where appropriate.1. The broker claims that the average time to sell a house is at most 48 days. a. State the Hypotheses that you are testing. (2)b. Using the data from the previous problem (page 2.), test the hypotheses (99% confidence level) using:(i) A test ratio (2)(ii) Critical values (2)(iii) A confidence interval (2)c. Find an approximate p-value for the null hypothesis. (1)d. Now find a 99% confidence interval for the standard deviation.(3)Solution: From Table 3 of the Syllabus Supplement:Interval for Confidence IntervalHypotheses Test Ratio Critical ValueMean ( Known)xzx20100:H:Hxxz0xcvzx20Mean ( Unknown)xstx21nDF0100:H:Hxsxt0xcvstx20a. 48:H48:H10 747.3,01.,41,48401.10ttnDFnFrom the previous page: 6.67x and 1392.1251441.2758.736nssx.b. (i) Test Ratio: 6146.11392.12486.670xsxt. This is in the ‘accept’ regionbelow 3.747, so do not reject H0. (ii) Critical Value: Since this is a one-sided test, xcvstx0 49.45501392.12747.348 or 95.49. This means that we reject H0 if the sample mean is above 95.49. since 6.67xis belowthis critical value, do not reject H0. (iii) Confidence Interval: Since this is a one-sided test, xstx 49.456.671392.12747.36.67 or 11.22. This does not contradict 48:H0, because any mean in the range 22.11 to 48 satisfies both statements, so do not reject H0.c. From the t table t= 1.6146 is smaller than 132.2405.tbut is larger than 533.1410.t, so that, forthis 1-sided test, we can say 05.10. pvalue.d. From page 1 of the Syllabus Supplement: n s n s 1 122222122 .41 nDF and01.. Since 005.2 and 995.21 , look up 8603.1442005. and3 2070.042995.. So 2995.222005.211snsn or 2070.08.73648603.148.73642 or68.1423733.1982. Finally, taking square roots, 32.119083.14 .2/22/99 252y99122. A new broker opens in town and a sample of nine houses gives the following days to sale:Home Days 1 62 2 28 3 114 4 113 5 716 297 538 549 34Assuming that the distribution is not normal, a. Find a confidence level for the following interval for the median: 7134 . (3)b. Test the hypotheses that the median is at most 33 at the 10% level. (5)c. If we have a sample of 350 numbers in order and take the 15th from each end, what would the confidence level be? (3)d. (Extra credit) What numbers would we use if we wanted a 99% confidence interval for the median and had a sample of 350 numbers? (3)Solution:a. If we put the numbers in order, we get 28 29 34 53 54 62 71 113 114. Thus we want the 3rd number from the end. If 9nand 3k, we find from the binomial table (with 5.p), 1211 kxP 82032.08984.21221 xP. 72 xPxPb. From the outline:Hypotheses about a median Hypotheses about a proportionIf p is the proportion above0If p is the proportion
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