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Unformatted text preview: STAT 20: Lecture Section 2. Spring 2011 Solution to Homework #2 (26 marks total, no extra marks for turning in homework) Note: Whenever yoa pick an an option or choose from True/False,
any answer without an explanation will he treated as a random
guess and will not get any marks. Q1. Text Page 106 Q7 (6 marks: I for part, 1 for part b and 4 for part c)
a. [(650 x 600) + (600 X. 400)] / (500 + 500)
= [(6) x650 + (.4) X600] 2 630
For the men and the women together, the average Math SAT score was 630.
Note: The overall average is closer to 650 than 600. b. For the men and the women together, the SD of Math SAT scores will be more
than 125. When you put the men and the women together, the separation of their
averages causes the SD to increase. When combining the two lists into one list,
the scores at the lower end of the women’s scores and the scores at the higher end
of the men’s scores are farther apart from the combined average. Therefore the
SD increases. (Refer to the Giants~Hobbits example on Page 67 of the Reader). e. For any list X1 ..... X” (done in lecture) ix? : n[(SD(X)2 +(X )2]
Therefore for the men: 2X3 2 600[(125)2 + (650)2] = 262,875,000.
while for the women: 2 X ,2 = 400[(l25)2 + (600)2] :2 150,250,000.
i=1 so for 2111 1000 people: i Xf m 262,875,000 + 150,250,000 = 413,125,000 For all 1000 the variance = “Mean of the squares” m “square of the mean” SO the variance is (413,125,000/1000) —— (630)2 = 16,225 So the SD is 1/16,225 = 127.38 Q2. Text Page 135 Q2 (4 marks: 2 for each part a and b)
a. The correlation between the age of the car and its gasoline economy (miles per
gallon) would be negative because older cars get fewer miles per gallon. The
plot would Eook something like this (graph not necessary to get credit): Gasoline
Economy Age of car b. The correlation between gasoline economy and income of owner turns out to be
positive because richer people tend to own newer cars and have the money to
maintain their cars better. Therefore, higher income of owners is positively
correlated with better gasoline economy. Q3. Text Page 135 Q3 (3 marks: to receive credit, the linear link between the two
"variables must be made, either with expiicit aigebra or a clear picture) x: husband’s height y: wife’s height given: y =2 x —— (.OSX) 2: (l—.08)X 2: .92): Therefore, a_ll the points on the scatter diagram for height of wife vs. height of
husband will lie on a straight line with a positive slope of .92. This is sufﬁcient to say that the correlation is +1.
The scatter pIot would look like this: Wife’ 5
Height Husband’s Height Q4. (From homework, reproduced here in hold): (4 marks: 1 for correct xx, 1 for correct zy, 1 for correct area, i for correct percentile. Do
not double—penalize, deduct 1 for incorrect zX or zy but give points for correct steps) For a large group of ten year old boys the correlation between height and
weight is 0.4 with a football shaped scatter diagram. Predict the percentile for weight of a boy in this group who is at the 7 Sin
percentile of heights. Assume that for this group both heights and weights
follow normal carves. x: height y: weight (we are given that r = .4)
At the 75th percentiie: z, a .67
Use formula: 2),: r - 2X so 2),: (.4)(.67) 2 .27 The area under the normal curve from -.27 to .27 z 22%
So area below 2 = .27 is [22% + 1/2 (100% - 22%)] z 61%. Therefore, the prediction for the weight of a boy who is at the 75Eh percentile of
heights is the 6}.St percentile. Note: EXCEL can give more accurate answers: =normsinv(.75) is .6745
a3, = r.zx is more accurately .2698 and =norrnsdist(.2698) is .6063 (rounded to .61) Q5. (from homework, reproduced here in hold): (3 marks: to receive credit, the link between the two variables should be shown) A multiple choice quiz has 10 questions each with 5 options. The correct choice gets 4 points while one point is subtracted for every
incorrect choice. The possible scores range from -10 to 40. A scatter plot is
obtained for 200 students (all of whom attempt every one of the 10
questions) showing the number of questions a student got right versus their
total score on the quiz. What is the correlation? Let x = number of questions right, then (10 - X) is the number of questions wrong
Let T a total score T=4X+(«1)( 10mx)=4x —10+x=5x~ 10 So given it, we can predict T with certainty, using a straight line equation.
The correlation is exactly +1. There is a perfect linear association between the
number of questions a student got right and their total score. ...
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- Spring '11