What is the mean of the numbers 1, 2, 3, ..., 10 with frequencies {}^{9}C_{0}, {}^{9}C_{1}, {}^{9}C_{2}, ..., {}^{9}C_{9}, respectively?
- A. 1\cdot1\times2^{8}
- B. 1\cdot2\times7^{4}
- C. 5.5 ✓
- D. 0.55
Correct Answer: C. 5.5
Explanation
The mean is \overline{X} = \frac{\sum f_i x_i}{\sum f_i}. Here x_i = i+1 and f_i = {}^9C_i for i=0 to 9. \sum f_i = 2^9. The numerator is \sum_{i=0}^9 (i+1){}^9C_i = \sum i{}^9C_i + \sum {}^9C_i. Using \sum i{}^nC_i = n2^{n-1}, we get 9 \times 2^8 + 2^9 = 2^8(9+2) = 11 \times 2^8. Thus, \overline{X} = \frac{11 \times 2^8}{2^9} = \frac{11}{2} = 5.5.
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