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Unformatted text preview: THE UNIVERSITY or HONG KONG 07/08
DEPARTMENT or STATISTICS AND ACTUARIAL SCIENCE STAT0301 Elementary Statistical Methods
Assignment 4 (Do all. Hand in solutions to the three starred questions on or before 13.11.07 1. (a) The following system of linear equations is given: 3=—a—4b, 12=2a—b, 14=3a+b, 21=5g+3bt,"25=4a—3b. (1) Find the normalized solutions.
(ii) What is the minimum squared discrepancy (i.e., least squares discrepancy)? *(b) The following table gives the statures and step lengths (both in cm) of 10 ran—
domly selected males in the age group 2O~302 Stature (y): 162 178 170 163 166 156 158 149 157 163
Step (33) : 75 80 78 71 73 67 70 62 69 73 (i) Fit a regression line 17 = a + bzc to the data for the purpose of estimating 3;
given m. A detective measured the step length of an escaped suspect (who had been conﬁrmed to be about 25 years old) to be 74 cm. How tall was the
suspect likely to be? (ii) Convert the 10 data pairs into ranks. Calculate the Spearman’s rank corre—
lation. 2. The following table shows the numbers of supermarkets and the numbers of small
stores in seven districts A, B, C, D, E, F and G. (a) Fit a regression line 3) = a + by: to the data. How do you interpret b?
(b) Calculate the correlation coefﬁcient 7'. Give an interpretation. (C) It has recently been observed that an eighth district has 10 supermarkets, and a
ninth district has 1 supermarket. Estimate the number of small stores each has.
Which estimate is more reliable? Why? 1 (d) A survey on three other districts is conducted. However, only their summary
statistics are available, as follows: = 24, = 148, = 242, = 8150, = 1389. Incorporating this additional information, mod—
ify the regression line and the correlation coefﬁcient, based on the 10 data pairs. >"3. It is believed that a worker’s expenditure (y) for entertainment, gambling, etc.,
is directly related to his salary A random sample of 60 workers in a certain
industry provided information in the following tw0~way frequency table, where :1:
and y are expressed in 1000 dollars per month. Salary Expenditure 5—7 7—9 9—11 11—13 13—15 (a) Calculate the sample correlation coefﬁcient between at and y.
(b) Fit a regression line of y given :6 to the data.
(c) Fit a regression line of a: given y to the data. *4. The following (hypothetical) data give, for n = 61 trials of various automobiles
traveling at speed 20 (miles per hour), the stopping distances y (feet): 2,4,8,8 14,19,34 57,78
'17 17 22,29 64,84
8,9,11,13 18 29,34,47 54,68
5,5,13 19 30 60,67,101
8,14,17 20 48 77
11,19,21 21 39,42,55 85,107
15,18,27 24 56 79 14,16 25 33,48,56,59 138 16 26 39,41 110,134 (a) Fit a regression line 3] = a + 03: to the data.
(b) Find the correlation coefﬁcient between :0 and y. (c) A similar test for 39 trials yield the following summary statistics: 5: = 20, g = 40,
31 = 10, 52 = 35, 7" = 0.90. Combine all the 100 pairs of data. Reﬁt a regression
line of y on a: and recalculate the correlation coefﬁcient. ...
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 Spring '07
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