Question

σ² = (87 - 78)² + (46 - 78)² + (90 - 78)² + (78 - 78)² + (89 - 78)²/5 = (9)² - (32)² + (12)² + (0)² + (11)²/5 = 81 - 1,024 + 144 + 0 + 121/5 = -678/5 = -135.6
what is the first error she made in computing the variance?
○ emi failed to find the difference of 89 - 78 correctly.
○ emi divided by n - 1 instead of n.
○ emi evaluated (46 - 78)² as -(32)².
○ emi forgot to take the square root of -135.6.

Explanation:

To determine the first error, we analyze each step:

1. For \(89 - 78\), \(89-78 = 11\), so the first option is incorrect.
2. The formula for population variance is \(\sigma^{2}=\frac{\sum(x_{i}-\mu)^{2}}{N}\), and here \(N = 5\) (dividing by \(N\), not \(N - 1\)), so the second option is incorrect as the error is not here.
3. For \((46 - 78)^{2}\), \(46-78=-32\), and \((-32)^{2}=32^{2}\), but in the calculation, it is written as \(-(32)^{2}\), which is wrong. This is the first error.
4. Variance is \(\sigma^{2}\), and we don't take the square root for variance (standard deviation is square root of variance), and also variance can't be negative (since it's a sum of squares divided by a positive number), but the first error occurs earlier, so the fourth option is incorrect.

Answer:

Emi evaluated \((46 - 78)^{2}\) as \(-(32)^{2}\).