Question

mariela earned the following test scores in math:
95, 90, 100, 85, 80
mr. adams accidentally entered mariela’s next test score as 0. which of these statements are true? check all that apply.
□ before the error, the mean and the median were both 90.
□ before the error, the mean was 90 and the median was 100.
□ the error added an outlier to the data set.
□ after the error, the median remained the same.
□ after the error, the mean was 75 and the median was 87.5.
□ the mean was impacted by the error more than the median.
□ only the median was impacted by the error.

Explanation:

Step1: Calculate mean before error

Sum of scores: \(95 + 90 + 100 + 85 + 80 = 450\)
Number of scores: \(5\)
Mean: \(\frac{450}{5} = 90\)

Step2: Find median before error

Sort scores: \(80, 85, 90, 95, 100\)
Median (middle value): \(90\)

Step3: Analyze outlier after error

New score: \(0\), which is far from other scores (80 - 100), so it's an outlier.

Step4: Check median after error

New data: \(0, 80, 85, 90, 95, 100\)
Median (average of two middle values): \(\frac{85 + 90}{2} = 87.5\)? Wait, no—wait, original median was 90 (odd count). After adding 0, count is 6 (even). Middle two: 85 and 90? Wait, no, sorted new data: \(0, 80, 85, 90, 95, 100\). Middle two are 85 and 90? Wait, no, position 3 and 4: 85 and 90? Wait, no, 6 numbers: positions 3 and 4. Wait, 0 (1), 80 (2), 85 (3), 90 (4), 95 (5), 100 (6). So middle two: 85 (3) and 90 (4). Median: \(\frac{85 + 90}{2} = 87.5\). Wait, but the option says "After the error, the median remained the same"—no, that's false? Wait, no—wait, maybe I made a mistake. Wait, original median was 90. After error, median is 87.5? But wait, let's recheck. Wait, no, maybe I messed up the sorted order. Wait, original scores: 80,85,90,95,100. After adding 0: 0,80,85,90,95,100. So middle two are 85 and 90? Wait, no, index 2 and 3 (0-based: 0,1,2,3,4,5). So elements at 2:85, 3:90. Median: (85+90)/2=87.5. But the option says "After the error, the median remained the same"—is that true? Wait, no, original median was 90, new is 87.5. So that option is false? Wait, but maybe I made a mistake. Wait, let's re-express:

Wait, the first option: "Before the error, the mean and the median were both 90"—true (steps 1-2).

Third option: "The error added an outlier"—true (0 is outlier).

Sixth option: "The mean was impacted by the error more than the median"—mean before:90, after: (450+0)/6=75 (change of 15). Median before:90, after:87.5 (change of 2.5). So mean impacted more—true.

Wait, let's check each option:

1. Before error: mean 90, median 90—true.

2. Before error: mean 90, median 100—false (median is 90).

3. Error added outlier—true (0 is outlier).

4. After error, median remained same—false (was 90, now 87.5).

5. After error, mean 75, median 87.5—mean: (450+0)/6=75, median: (85+90)/2=87.5—true? Wait, but earlier I thought median was 87.5, but the option says "After the error, the mean was 75 and the median was 87.5"—is that true? Let's check:

Mean after error: (95+90+100+85+80+0)/6 = 450/6 = 75—correct.

Median after error: sorted data 0,80,85,90,95,100. Middle two: 85 and 90. Median: (85+90)/2 = 87.5—correct. So this option is true?

Wait, but earlier step 4: "After the error, the median remained the same"—no, it changed from 90 to 87.5. So that option is false.

6. Mean impacted more than median—true (mean changed by 15, median by 2.5).

7. Only median impacted—false (mean also impacted).

So true options:

- Before the error, the mean and the median were both 90.

- The error added an outlier to the data set.

- After the error, the mean was 75 and the median was 87.5.

- The mean was impacted by the error more than the median.

Wait, but let's recheck the median after error. Wait, original data: 5 numbers, median 90 (3rd term). After adding 0, 6 numbers: 3rd and 4th terms are 85 and 90. So median is (85+90)/2 = 87.5, which is different from 90. So the option "After the error, the median remained the same" is false.

Now, let's confirm each option:

1. True (mean 90, median 90).

2. False.

3. True (0 is outlier).

4. False (median changed).

5. True (mean 75, median 87.5).

6. True (mean change: 90→75 (15), median: 90→87.5 (2.5)).

7. False (mean also impacted).

So the true statements are:

- Before the error, the mean and the median were both 90.

- The error added an outlier to the data set.

- After the error, the mean was 75 and the median was 87.5.

- The mean was impacted by the error more than the median.

But let's check the first option again:

Before error: scores 95,90,100,85,80. Sum is 95+90=185, +100=285, +85=370, +80=450. Mean 450/5=90. Sorted: 80,85,90,95,100. Median is 90. So first option is true.

Third option: 0 is an outlier (since all other scores are 80-100, 0 is far below), so true.

Fifth option: mean after error is (450+0)/6=75, median is (85+90)/2=87.5, so true.

Sixth option: mean changed by 15 (90→75), median by 2.5 (90→87.5), so mean impacted more, true.

Wait, but the problem says "Check all that apply". So the correct options are:

- Before the error, the mean and the median were both 90.

- The error added an outlier to the data set.

- After the error, the mean was 75 and the median was 87.5.

- The mean was impacted by the error more than the median.

But let's check the options again:

Options:

1. Before the error, the mean and the median were both 90. → True.

2. Before the error, the mean was 90 and the median was 100. → False.

3. The error added an outlier to the data set. → True.

4. After the error, the median remained the same. → False.

5. After the error, the mean was 75 and the median was 87.5. → True.

6. The mean was impacted by the error more than the median. → True.

7. Only the median was impacted by the error. → False.

So the true options are 1, 3, 5, 6.

But let's confirm the median after error again. Original median: 90 (5 numbers, middle is 3rd: 90). After adding 0, 6 numbers: middle two are 3rd (85) and 4th (90). So median is (85+90)/2=87.5, which is different from 90, so option 4 is false. Option 5 says median is 87.5, which is correct.

So the correct answers are:

- Before the error, the mean and the median were both 90.

- The error added an outlier to the data set.

- After the error, the mean was 75 and the median was 87.5.

- The mean was impacted by the error more than the median.

Answer:

• Before the error, the mean and the median were both 90.
• The error added an outlier to the data set.
• After the error, the mean was 75 and the median was 87.5.
• The mean was impacted by the error more than the median.