A sample has a mean of M = 86. If one new person is added to the sample, and Σ X is unchanged, what effect will the addition have on the sample mean?Group of answer choicesThe sample mean will decrease.The sample mean will increase a little.The sample mean will increase substantially.The sample mean will remain the same.
Question
A sample has a mean of M = 86. If one new person is added to the sample, and Σ X is unchanged, what effect will the addition have on the sample mean?Group of answer choicesThe sample mean will decrease.The sample mean will increase a little.The sample mean will increase substantially.The sample mean will remain the same.
Solution
To determine the effect of adding one new person to the sample while keeping Σ X unchanged, follow these steps:
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Understand the given information:
- The current mean of the sample is M = 86.
- The sum of all scores in the sample is Σ X.
- A new person is added to the sample, but Σ X remains the same.
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Recall the formula for the mean:
- Mean (M) = Σ X / N, where N is the number of individuals in the sample.
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Analyze the situation:
- Initially, the mean is 86, so Σ X = 86 * N.
- After adding one new person, the number of individuals becomes N + 1.
- Since Σ X is unchanged, the new mean (M') will be Σ X / (N + 1).
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Calculate the new mean:
- New mean (M') = Σ X / (N + 1) = (86 * N) / (N + 1).
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Compare the new mean to the original mean:
- Since N + 1 is greater than N, dividing the same Σ X by a larger number will result in a smaller mean.
Therefore, the sample mean will decrease. The correct answer is:
- The sample mean will decrease.
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