Find expert answers and community support for all your questions on IDNLearn.com. Join our interactive Q&A community and access a wealth of reliable answers to your most pressing questions.
Sagot :
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.
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Your questions deserve accurate answers. Thank you for visiting IDNLearn.com, and see you again for more solutions.
