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A study guide for exam i of math 102 / core 143 cx, including concepts related to histograms, averages, standard deviations, correlation, normal approximation, and experimental design. It includes sample problems and solutions to help students prepare for the exam.
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October 3, 2006
Exam I — Math 102 / Core 143 CX
Points are in parentheses. Show your work to receive partial credit; an answer like [1 − (6/7)^3 ]/ 4 is worth more than 0.09257, because it displays your reasoning more clearly.
(a) Years of education from a large group of Americans (b) Numbers on the caller’s markers from an old bingo game, missing several markers (c) Heights of a large group of nine-year-old boys (d) Numbers of ways to get a given number on a roll of a pair of dice
2, Find, for the list of 10 numbers
− 4 , − 1 , − 1 , − 1 , 0 , 0 , 0 , 2 , 2 , 3 :
(a) the average (b) the standard deviation (c) the median
(d) the 35th percentile (e) the IQR
(a) Is the sign of the correlation reasonable? Explain. (b) Estimate the gas price when the oil production is 6 Mbbl/day. (c) About how far should you expect your estimate in (b) to be off? (d) If you were to estimate the oil production when the gas price is 2.3, would your estimate be closest to 1, 4, 6 or 9?
(c) The list of 20 numbers
− 14 , − 13 , − 12 , − 11 , − 10 , − 9 , − 8 , − 7 , − 6 , − 5 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14
has an average of 0 (that should be clear from the symmetry) and a standard deviation of not quite 10; so only 50% of the data is within one standard deviation of the average. How does this relate to the “Rule of Thumb” in the text? (d) For the data in (c), without doing arithmetic, would you guess that the standard devi- ations of the shorter lists
− 14 , − 13 , − 12 , − 11 , − 10 , − 9 , − 8 , − 7 , − 6 , − 5 and 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14
are equal to, less than, or greater than the standard deviation in (c)?
(a) Put a questionnaire on the web asking for responses from those who have taken vitamin C for cold symptoms. (b) Get a list of people who have purchased vitamin C, and interview as many as possible, including in your study the ones who say they took it for cold symptoms. (c) Get a list of cold sufferers, give them vitamin C and interview them for results. (d) Get a list of cold sufferers, give half of them (chosen at random) vitamin C, and interview all of them for results.
Some possibly useful formulas:
s =
√ ∑ (x − x)^2 n zx =
x − x s r =
∑ zxzy n
RMS error = σy
√ 1 − r^2 y − y = r
sy sx (x − x) y − y = (sign of r)
sy sx (x − x)
Solutions to Exam I
√ [(− 4 − 0)^2 + 3(− 1 − 0)^2 + 3(0 − 0)^2 + 2(2 − 0)^2 + (3 − 0)^2 ]/ 10 ≈ 1. 9 (c) 0, between the fifth and sixth numbers in the list (d) −1, between the third and fourth numbers in the list (e) The third quartile may be 0 or 2 or anything between, but the first quartile is surely −1, so their difference, the IQR, could be anything from 1 to 3.
√ 1 − (.5)^2 ≈
. 69 (d) A gas price of 2.3 is one SD above the average, so we should expect oil production to be .5 SD’s below average, or 5−(.5)(2) = 4 (in other words, below average, but not as much below average as the gas price was above its average, both measured in standard units).