Answer :
Yuri's first error occurred when he divided by the total number of data points [tex]\( n \)[/tex] instead of [tex]\( n - 1 \)[/tex]. This is a crucial step when calculating the sample standard deviation. For a sample, the denominator should be [tex]\( n - 1 \)[/tex] to account for the degrees of freedom in the estimation of the population standard deviation from the sample data.
Let's go through the steps correctly:
1. Calculate the mean:
[tex]\[ \text{mean} = \frac{12 + 14 + 9 + 21}{4} = \frac{56}{4} = 14 \][/tex]
2. Find the squared differences from the mean:
[tex]\[ (12 - 14)^2 = (-2)^2 = 4 \][/tex]
[tex]\[ (14 - 14)^2 = 0^2 = 0 \][/tex]
[tex]\[ (9 - 14)^2 = (-5)^2 = 25 \][/tex]
[tex]\[ (21 - 14)^2 = 7^2 = 49 \][/tex]
3. Sum the squared differences:
[tex]\[ 4 + 0 + 25 + 49 = 78 \][/tex]
4. Divide by [tex]\( n - 1 \)[/tex] (since [tex]\( n = 4 \)[/tex]):
[tex]\[ \frac{78}{4 - 1} = \frac{78}{3} = 26 \][/tex]
5. Take the square root to find the sample standard deviation:
[tex]\[ s = \sqrt{26} \][/tex]
Therefore, the correct sample standard deviation is:
[tex]\[ s = \sqrt{26} \][/tex]
The first error Yuri made was not dividing by [tex]\( n - 1 \)[/tex] (which is 3 in this case) when calculating the sample standard deviation. Instead, he incorrectly divided by [tex]\( n \)[/tex].
Let's go through the steps correctly:
1. Calculate the mean:
[tex]\[ \text{mean} = \frac{12 + 14 + 9 + 21}{4} = \frac{56}{4} = 14 \][/tex]
2. Find the squared differences from the mean:
[tex]\[ (12 - 14)^2 = (-2)^2 = 4 \][/tex]
[tex]\[ (14 - 14)^2 = 0^2 = 0 \][/tex]
[tex]\[ (9 - 14)^2 = (-5)^2 = 25 \][/tex]
[tex]\[ (21 - 14)^2 = 7^2 = 49 \][/tex]
3. Sum the squared differences:
[tex]\[ 4 + 0 + 25 + 49 = 78 \][/tex]
4. Divide by [tex]\( n - 1 \)[/tex] (since [tex]\( n = 4 \)[/tex]):
[tex]\[ \frac{78}{4 - 1} = \frac{78}{3} = 26 \][/tex]
5. Take the square root to find the sample standard deviation:
[tex]\[ s = \sqrt{26} \][/tex]
Therefore, the correct sample standard deviation is:
[tex]\[ s = \sqrt{26} \][/tex]
The first error Yuri made was not dividing by [tex]\( n - 1 \)[/tex] (which is 3 in this case) when calculating the sample standard deviation. Instead, he incorrectly divided by [tex]\( n \)[/tex].
