IDNLearn.com: Where curiosity meets clarity and questions find their answers. Get the information you need from our experts, who provide reliable and detailed answers to all your questions.
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.
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for choosing IDNLearn.com. We’re here to provide reliable answers, so please visit us again for more solutions.