Solution
Back-up Theory
Properties of Mean
If xi is the same for all i = 1 to n, say x, then mean = nx……………………………...........................................…............….. (1)
If Xbar is the mean of X based on n1 sample observations and Ybar is the mean of Y based on n2 sample observations,
then the combined mean, say M = (n1Xbar + n2Ybar)/(n1 + n2) ….....................................................................................…. (2)
Properties of Variance
If xi is the same for all i = 1 to n, say x, then variance = 0 …………….....................................................…………………….. (3)
If s12 is the variance of X based on n1 sample observations on X and s22 is the variance of Y based on n2 sample observations on Y, then the combined variance, say s2= {(n1- 1)s12 + (n2 - 1)s22}/(n1 + n2 - 2) ………………………….…. (4)
Now to work out the solution,
Given n1 = 149, Xbar = 1809 [vide (1)], s12 = 0 [vide (3)], n2 = 109, Ybar = 1879, s22 = 1361788,
Part (a)
Vide (2), combined average (mean) = {(149 x 1809) + (109 x 1879)}/(149 + 109)
= $1838.6 Answer
Part (b)
In the combined sample, there are 258 values. So, the combined sample median would be the average of 129th and 130th values in the ordered set of the combined values.
Case 1 No value in the second sample is less than 1809. Then, the ordered set would be:
1809, 1809, ….., 1809, (149 values), followed by 109 values of the second sample and hence both 129th and 130th values in the ordered set of the combined values would be 1809. Thus, the median = 1809
Case 2 m values in the second sample are less than 1809.
Since mean of the second sample is 1879, at least one value must be greater than 1879 and hence greater than 1809. So, m < 109, say 108 as an extreme case. Then, the ordered set would be:
108 values of the second sample, 1809, 1809, ….., 1809, (149 values), followed by 1 value of the second sample and hence both 129th and 130th values in the ordered set of the combined values would be 1809. Thus, the median = 1809.
Consequently, the median of the combined sample values is always 1809 Answer
Part (c)
Since, as brought out in the above analyses, median > mean the distribution would be
positively skewed, i.e., longer right tail. Answer
Part (d)
Vide (4), combined variance = {(148 x 0) + (108 x 1361788}/(148 + 108)
= 574504
So, combined standard deviation = sqrt(574504)
= $756.96 Answer
DONE
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