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Solutions to exam ii for math 102 / core 143, covering topics such as conditional probability, normal distribution, and central limit theorem.
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November 18, 2008 Math 102 / Core 143 CX — Exam II
Put your answers to all the problems on the numbered sheet, except for the essay (#7), which goes on the back. The blue book is for your scratch work and will NOT be graded.
(a) The average value of the cards chosen is at least 4. (b) The average value of the cards chosen is at least 8. (c) The sum of the cards chosen it at least 8 more than 5 times the number of cards chosen. (d) The sum of the cards chosen is at least 8 times the number of cards chosen. (e) More than 20% of the cards chosen are 6’s (exactly). (Warning: This should change your “box model” from the one you were using in (a) through (d).)
(a) Find a 90% confidence interval for the average distance from a household in the city to the nearest grocery. (b) How many households must be surveyed in order to get a 90% confidence interval only a third as wide as the one you found in (a) (assuming the new survey has the same x and s as this one.)
Some possibly useful formulas:
n! k!(n − k)! pk(1 − p)n−k
√ total population − sample size total population − 1
σ = (larger − smaller)
√ (fraction with larger)(fraction with smaller) =
√ average of (x − x)^2
EV of sum (or count) = n · (AV of box) SE of sum (or count) =
n · (SD of box)
EV of avg (or %) = AV of box SE of avg (or %) = SD of box √ n Normal table (Area between −z and z) z Area(%) z Area(%) z Area(%) z Area(%) z Area(%) 0.0 0.0 0.9 63.19 1.8 92.81 2.7 99.31 3.6 99. 0.05 3.99 0.95 65.79 1.85 93.57 2.75 99.4 3.65 99. 0.1 7.97 1 68.27 1.9 94.26 2.8 99.49 3.7 99. 0.15 11.92 1.05 70.63 1.95 94.88 2.85 99.56 3.75 99. 0.2 15.85 1.1 72.87 2 95.45 2.9 99.63 3.8 99. 0.25 19.74 1.15 74.99 2.05 95.96 2.95 99.68 3.85 99. 0.3 23.58 1.2 76.99 2.1 96.43 3 99.73 3.9 99. 0.35 27.37 1.25 78.87 2.15 96.84 3.05 99.771 3.95 99. 0.4 31.08 1.3 80.64 2.2 97.22 3.1 99.806 4 99. 0.45 34.73 1.35 82.3 2.25 97.56 3.15 99.837 4.05 99. 0.5 38.29 1.4 83.85 2.3 97.86 3.2 99.863 4.1 99. 0.55 41.77 1.45 85.29 2.35 98.12 3.25 99.885 4.15 99. 0.6 45.15 1.5 86.64 2.4 98.36 3.3 99.903 4.2 99. 0.65 48.43 1.55 87.89 2.45 98.57 3.35 99.919 4.25 99. 0.7 51.61 1.6 89.04 2.5 98.76 3.4 99.933 4.3 99. 0.75 54.67 1.65 90.11 2.55 98.92 3.45 99.944 4.35 99. 0.8 57.63 1.7 91.09 2.6 99.07 3.5 99.953 4.4 99. 0.85 60.47 1.75 91.99 2.65 99.2 3.55 99.961 4.45 99.
Math 102 / Core 143 — Solutions to Exam II
100 = .577, so, using the normal table,
P (9 < avg < 12) = P
( 9 − 10. 5
. 577 < z <
) = P (− 2. 6 < z < 2 .6) ≈. 99.