Let's break down the problem and find the inaccuracies in Emi's process.
First, let's identify the correct steps for computing the variance of a dataset:
1. Calculate the mean of the dataset. This is given as 78.
2. Find the differences between each data point and the mean.
- For 87: [tex]\( 87 - 78 = 9 \)[/tex]
- For 46: [tex]\( 46 - 78 = -32 \)[/tex]
- For 90: [tex]\( 90 - 78 = 12 \)[/tex]
- For 78: [tex]\( 78 - 78 = 0 \)[/tex]
- For 89: [tex]\( 89 - 78 = 11 \)[/tex]
3. Square each of these differences.
- For 87: [tex]\( 9^2 = 81 \)[/tex]
- For 46: [tex]\( (-32)^2 = 1024 \)[/tex]
- For 90: [tex]\( 12^2 = 144 \)[/tex]
- For 78: [tex]\( 0^2 = 0 \)[/tex]
- For 89: [tex]\( 11^2 = 121 \)[/tex]
Therefore, the squared differences are:
[tex]\[ [81, 1024, 144, 0, 121] \][/tex]
4. Sum these squared differences.
[tex]\[ 81 + 1024 + 144 + 0 + 121 = 1370 \][/tex]
5. Divide this sum by the number of data points (N = 5), as this is a population variance calculation.
[tex]\[ \frac{1370}{5} = 274 \][/tex]
So, the correct variance should be 274.
Now identify the errors in Emi's calculation:
Let's review the steps Emi took:
- Emi calculated the differences and squared them, which she actually got correct as: [tex]\( [81, 1024, 144, 0, 121] \)[/tex].
- The key step before misdividing is adding up these squared differences: [tex]\(81 + 1024 + 144 + 0 + 121 = 1370\)[/tex].
Despite these correct steps, Emi misrepresented the addition step:
[tex]\[ \frac{81-1024+144+0+121}{5} = \frac{-678}{5} \][/tex]
The mistake here is:
1. Emi inaccurately calculated: [tex]\(81 - 1024 + 144 + 0 + 121 \)[/tex]. Instead, it should be [tex]\(81 + 1024 + 144 + 0 + 121\)[/tex].
Therefore, the first mistake Emi made is incorrect addition of squared differences. Option presented specific errors are:
- "Emi failed to find the difference of 89 - 78 correctly," which seems correct since differences listing seems correct.
- "Emi divided by [tex]\(N-1\)[/tex] instead of [tex]\(N\)[/tex]," which implies different statistics logic, but her aggregation misled more immediate fault during squared sum first inaccuracy pinpoint.
Thus, ultimately, identification clears on Emi's primary fault in wrongly summed squared differences accurately leading correct analysis.
