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Sagot :
Let's go through the given steps to identify where Yuri made an error in computing the standard deviation.
### Step-by-Step Solution
1. Compute the Mean:
The given data set is [tex]\([12, 14, 9, 21]\)[/tex].
- The mean [tex]\(\mu\)[/tex] is already given as 14. This is correct.
[tex]\[ \mu = 14 \][/tex]
2. Calculate the Squared Differences from the Mean:
We need to find the squared differences of each data point from the mean:
- For [tex]\(12\)[/tex]: [tex]\((12 - 14)^2 = (-2)^2 = 4\)[/tex]
- For [tex]\(14\)[/tex]: [tex]\((14 - 14)^2 = (0)^2 = 0\)[/tex]
- For [tex]\(9\)[/tex]: [tex]\((9 - 14)^2 = (-5)^2 = 25\)[/tex]
- For [tex]\(21\)[/tex]: [tex]\((21 - 14)^2 = (7)^2 = 49\)[/tex]
[tex]\[ \text{Squared Differences} = [4, 0, 25, 49] \][/tex]
3. Sum of Squared Differences:
We then sum these squared differences:
[tex]\[ 4 + 0 + 25 + 49 = 78 \][/tex]
- The sum of the squared differences is correctly calculated as 78.
4. Calculate the Variance:
To find the variance for a sample, we need to divide the sum of squared differences by [tex]\(n-1\)[/tex] (degrees of freedom correction), where [tex]\(n\)[/tex] is the sample size.
[tex]\[ \text{Sample size} = 4 \][/tex]
[tex]\[ \text{Variance} = \frac{78}{4-1} = \frac{78}{3} = 26 \][/tex]
5. Calculate the Correct Standard Deviation:
The standard deviation is the square root of the variance.
[tex]\[ \text{Standard Deviation} = \sqrt{26} \approx 5.099 \][/tex]
### Error Identification
Yuri computed the standard deviation using the division by [tex]\(n\)[/tex] instead of [tex]\(n-1\)[/tex]. Let's verify this calculation:
[tex]\[ \sqrt{\frac{78}{4}} = \sqrt{19.5} \approx 4.416 \][/tex]
The error Yuri made was in failing to use [tex]\(n-1\)[/tex] when computing the standard deviation for the sample data. Instead of using the corrected formula for variance, which divides the sum of squared differences by [tex]\(n-1\)[/tex], he incorrectly divided by [tex]\(n\)[/tex]. The correct computation should divide by 3 (since 4-1=3), not 4.
### Conclusion
The first error Yuri made in computing the standard deviation was failing to use [tex]\(n-1\)[/tex] (degrees of freedom correction). Instead, he incorrectly used [tex]\(n\)[/tex] in the calculation of the variance. Thus, the correct approach to compute the standard deviation for a sample is to divide the sum of squared differences by [tex]\(n-1\)[/tex] and then take the square root of that value.
### Step-by-Step Solution
1. Compute the Mean:
The given data set is [tex]\([12, 14, 9, 21]\)[/tex].
- The mean [tex]\(\mu\)[/tex] is already given as 14. This is correct.
[tex]\[ \mu = 14 \][/tex]
2. Calculate the Squared Differences from the Mean:
We need to find the squared differences of each data point from the mean:
- For [tex]\(12\)[/tex]: [tex]\((12 - 14)^2 = (-2)^2 = 4\)[/tex]
- For [tex]\(14\)[/tex]: [tex]\((14 - 14)^2 = (0)^2 = 0\)[/tex]
- For [tex]\(9\)[/tex]: [tex]\((9 - 14)^2 = (-5)^2 = 25\)[/tex]
- For [tex]\(21\)[/tex]: [tex]\((21 - 14)^2 = (7)^2 = 49\)[/tex]
[tex]\[ \text{Squared Differences} = [4, 0, 25, 49] \][/tex]
3. Sum of Squared Differences:
We then sum these squared differences:
[tex]\[ 4 + 0 + 25 + 49 = 78 \][/tex]
- The sum of the squared differences is correctly calculated as 78.
4. Calculate the Variance:
To find the variance for a sample, we need to divide the sum of squared differences by [tex]\(n-1\)[/tex] (degrees of freedom correction), where [tex]\(n\)[/tex] is the sample size.
[tex]\[ \text{Sample size} = 4 \][/tex]
[tex]\[ \text{Variance} = \frac{78}{4-1} = \frac{78}{3} = 26 \][/tex]
5. Calculate the Correct Standard Deviation:
The standard deviation is the square root of the variance.
[tex]\[ \text{Standard Deviation} = \sqrt{26} \approx 5.099 \][/tex]
### Error Identification
Yuri computed the standard deviation using the division by [tex]\(n\)[/tex] instead of [tex]\(n-1\)[/tex]. Let's verify this calculation:
[tex]\[ \sqrt{\frac{78}{4}} = \sqrt{19.5} \approx 4.416 \][/tex]
The error Yuri made was in failing to use [tex]\(n-1\)[/tex] when computing the standard deviation for the sample data. Instead of using the corrected formula for variance, which divides the sum of squared differences by [tex]\(n-1\)[/tex], he incorrectly divided by [tex]\(n\)[/tex]. The correct computation should divide by 3 (since 4-1=3), not 4.
### Conclusion
The first error Yuri made in computing the standard deviation was failing to use [tex]\(n-1\)[/tex] (degrees of freedom correction). Instead, he incorrectly used [tex]\(n\)[/tex] in the calculation of the variance. Thus, the correct approach to compute the standard deviation for a sample is to divide the sum of squared differences by [tex]\(n-1\)[/tex] and then take the square root of that value.
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