Stab22 Midterm Examination Solutions - University Of Toronto Scarborough, 2007 Page 5
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117. (8 marks) A set of exam marks has mean 70, median 65, inter-quartile range 25 and
SD 15 marks. It is decided to subtract 10 from all the marks. For the new set of marks,
(a) what is the mean?
(b) what is the median?
(c) what is the inter-quartile range?
(d) what is the SD?
Mean and median are the original values minus 10 (60 and 55), IQR and
SD are measures of spread so they are unchanged (25 and 15).
8. (2 marks) A set of data has quartiles 30 and 75, and median 40. What would you
conclude about the shape of the data distribution?
Skewed to the right (because the median is closer to Q1 than Q3).
9. (12 marks) For a particular group of adult males, the distribution of cholesterol readings
is normal with mean 210 and SD 15. Use Table A to find the following, showing your
calculation in each case:
(a) The proportion of males in this group with cholesterol reading less than 240.
For 240, z = (240 − 210)/15 = 2; proportion less (from table) is
0.9772.
(b) The proportion of males in this group with cholesterol readings between 200 and
240.
For 200, z = (200 − 210)/15 = −0.67; proportion less (from ta-
ble)=0.2514. Proportion less than 240 (from (a)) is 0.9772, so proportion
between is 0.9772 − 0.2514 = 0.7258.
(c) The cholesterol reading that 20% of males in this group are higher than.
This was admittedly not stated as clearly as it might have been. An-
other way to ask it is: “Suppose x is a cholesterol reading, and 20% of
males have higher cholesterol reading than x. What is x?”
Table gives you less than, so 20% higher means 80% lower. Look up
0.8000 backwards in the body of the table, to get z = 0.84 (closest).
Then the required level is (0.84)(15) + 210 = 222.6. (Or solve 0.84 =
(x − 210)/15 for x if you prefer that way.)
(d) The first quartile of cholesterol readings for males in this group.
Like (c), but now we want 25% lower (that’s what the first quartile
is). Looking up 0.2500 in the body of the table gives z = −0.67 approx-
imately. Thus the first quartile is (−0.67)(15) + 210 = 200. (Compare
the first part of (b).)
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