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To solve the given problem and determine the mean, variance, and standard deviation of the final exam scores, we will go through the steps methodically.
### Step 1: Calculate the Mean
First, we need to determine the mean [tex]\(\mu\)[/tex] of the given scores. The mean is calculated by summing all the scores and then dividing by the number of scores.
Given scores: [tex]\(90, 85, 74, 91, 89, 66\)[/tex]
[tex]\[ \mu = \frac{90 + 85 + 74 + 91 + 89 + 66}{6} = \frac{495}{6} = 82.5 \][/tex]
So, the mean [tex]\(\mu\)[/tex] is 82.5.
### Step 2: Create the Table
Next, we create a table where we calculate the deviation from the mean ([tex]\(x - \mu\)[/tex]) for each score [tex]\(x\)[/tex], and then we square those deviations [tex]\((x - \mu)^2\)[/tex].
| [tex]\(x\)[/tex] | [tex]\(x - \mu\)[/tex] | [tex]\((x - \mu)^2\)[/tex] |
|-------|-------------|------------------|
| 90 | 90 - 82.5 = 7.5 | [tex]\(7.5^2 = 56.25\)[/tex] |
| 85 | 85 - 82.5 = 2.5 | [tex]\(2.5^2 = 6.25\)[/tex] |
| 74 | 74 - 82.5 = -8.5 | [tex]\((-8.5)^2 = 72.25\)[/tex] |
| 91 | 91 - 82.5 = 8.5 | [tex]\(8.5^2 = 72.25\)[/tex] |
| 89 | 89 - 82.5 = 6.5 | [tex]\(6.5^2 = 42.25\)[/tex] |
| 66 | 66 - 82.5 = -16.5| [tex]\((-16.5)^2 = 272.25\)[/tex] |
Thus, we have the following completed table:
| [tex]\(x\)[/tex] | [tex]\(x - \mu\)[/tex] | [tex]\((x - \mu)^2\)[/tex] |
|-------|-------------|------------------|
| 90 | 7.5 | 56.25 |
| 85 | 2.5 | 6.25 |
| 74 | -8.5 | 72.25 |
| 91 | 8.5 | 72.25 |
| 89 | 6.5 | 42.25 |
| 66 | -16.5 | 272.25 |
### Step 3: Calculate the Variance
The variance [tex]\(\sigma^2\)[/tex] is the average of the squared differences from the mean. Since we are treating the data as a population, the variance is calculated as:
[tex]\[ \sigma^2 = \frac{\sum{(x - \mu)^2}}{N} \][/tex]
Where [tex]\(N\)[/tex] is the number of scores.
[tex]\[ \sigma^2 = \frac{56.25 + 6.25 + 72.25 + 72.25 + 42.25 + 272.25}{6} = \frac{521.5}{6} = 86.91666666666667 \][/tex]
So, the variance [tex]\(\sigma^2\)[/tex] is approximately 86.92.
### Step 4: Calculate the Standard Deviation
The standard deviation [tex]\(\sigma\)[/tex] is the square root of the variance:
[tex]\[ \sigma = \sqrt{\sigma^2} = \sqrt{86.91666666666667} = 9.322910847297997 \][/tex]
So, the standard deviation [tex]\(\sigma\)[/tex] is approximately 9.32.
### Summary
- Mean: 82.5
- Variance: 86.92
- Standard Deviation: 9.32
The completed table and all required calculations are presented above.
### Step 1: Calculate the Mean
First, we need to determine the mean [tex]\(\mu\)[/tex] of the given scores. The mean is calculated by summing all the scores and then dividing by the number of scores.
Given scores: [tex]\(90, 85, 74, 91, 89, 66\)[/tex]
[tex]\[ \mu = \frac{90 + 85 + 74 + 91 + 89 + 66}{6} = \frac{495}{6} = 82.5 \][/tex]
So, the mean [tex]\(\mu\)[/tex] is 82.5.
### Step 2: Create the Table
Next, we create a table where we calculate the deviation from the mean ([tex]\(x - \mu\)[/tex]) for each score [tex]\(x\)[/tex], and then we square those deviations [tex]\((x - \mu)^2\)[/tex].
| [tex]\(x\)[/tex] | [tex]\(x - \mu\)[/tex] | [tex]\((x - \mu)^2\)[/tex] |
|-------|-------------|------------------|
| 90 | 90 - 82.5 = 7.5 | [tex]\(7.5^2 = 56.25\)[/tex] |
| 85 | 85 - 82.5 = 2.5 | [tex]\(2.5^2 = 6.25\)[/tex] |
| 74 | 74 - 82.5 = -8.5 | [tex]\((-8.5)^2 = 72.25\)[/tex] |
| 91 | 91 - 82.5 = 8.5 | [tex]\(8.5^2 = 72.25\)[/tex] |
| 89 | 89 - 82.5 = 6.5 | [tex]\(6.5^2 = 42.25\)[/tex] |
| 66 | 66 - 82.5 = -16.5| [tex]\((-16.5)^2 = 272.25\)[/tex] |
Thus, we have the following completed table:
| [tex]\(x\)[/tex] | [tex]\(x - \mu\)[/tex] | [tex]\((x - \mu)^2\)[/tex] |
|-------|-------------|------------------|
| 90 | 7.5 | 56.25 |
| 85 | 2.5 | 6.25 |
| 74 | -8.5 | 72.25 |
| 91 | 8.5 | 72.25 |
| 89 | 6.5 | 42.25 |
| 66 | -16.5 | 272.25 |
### Step 3: Calculate the Variance
The variance [tex]\(\sigma^2\)[/tex] is the average of the squared differences from the mean. Since we are treating the data as a population, the variance is calculated as:
[tex]\[ \sigma^2 = \frac{\sum{(x - \mu)^2}}{N} \][/tex]
Where [tex]\(N\)[/tex] is the number of scores.
[tex]\[ \sigma^2 = \frac{56.25 + 6.25 + 72.25 + 72.25 + 42.25 + 272.25}{6} = \frac{521.5}{6} = 86.91666666666667 \][/tex]
So, the variance [tex]\(\sigma^2\)[/tex] is approximately 86.92.
### Step 4: Calculate the Standard Deviation
The standard deviation [tex]\(\sigma\)[/tex] is the square root of the variance:
[tex]\[ \sigma = \sqrt{\sigma^2} = \sqrt{86.91666666666667} = 9.322910847297997 \][/tex]
So, the standard deviation [tex]\(\sigma\)[/tex] is approximately 9.32.
### Summary
- Mean: 82.5
- Variance: 86.92
- Standard Deviation: 9.32
The completed table and all required calculations are presented above.
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