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Two data samples are combined into one group. The first sample has 100 items with mean 15 and standard deviation 3. The combined group has 250 items with mean 15.6 and standard deviation sqrt(13.44). Which of the following statements is/are correct?

1. The mean of the second group is 16
2. The mean of the second group is 18
3. The standard deviation of the second group is 4
4. The standard deviation of the second group is 5

Correct answer: The mean of the second group is 16

Solution

Second group size n2 = 250 - 100 = 150. Combined mean: (100*15 + 150*x2_bar)/250 = 15.6 => 1500 + 150*x2_bar = 3900 => x2_bar = 16. For standard deviation: using the combined variance formula with d1 = 15-15.6 = -0.6 and d2 = 16-15.6 = 0.4, combined variance = (100*(9 + 0.36) + 150*(sigma2² + 0.16))/250 = 13.44. Solving: 100*9.36 + 150*sigma2² + 24 = 3360 => 936 + 150*sigma2² + 24 = 3360 => sigma2² = 16 => sigma2 = 4.

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