Solved on Feb 19, 2024

Find the mean of a dataset with S2=10S^{2}=10, n=30n=30, and x2=3290\sum x^{2}=3290.

STEP 1

Assumptions
1. The sample variance S2S^2 is 10.
2. The sample size nn is 30.
3. The sum of the squares of the observations x2\sum x^2 is 3290.
4. We are asked to find the mean of the observations.

STEP 2

Recall the formula for the sample variance S2S^2:
S2=1n1(x2(x)2n)S^2 = \frac{1}{n-1} \left(\sum x^2 - \frac{(\sum x)^2}{n}\right)

STEP 3

We can rearrange the formula to solve for x\sum x (the sum of the observations), which is necessary to find the mean:
S2=1n1(x2(x)2n)(n1)S2=x2(x)2nS^2 = \frac{1}{n-1} \left(\sum x^2 - \frac{(\sum x)^2}{n}\right) \Rightarrow (n-1)S^2 = \sum x^2 - \frac{(\sum x)^2}{n}

STEP 4

Multiply both sides by nn to clear the fraction:
n(n1)S2=nx2(x)2n(n-1)S^2 = n\sum x^2 - (\sum x)^2

STEP 5

Plug in the given values for S2S^2, nn, and x2\sum x^2 into the equation:
30(301)10=303290(x)230(30-1) \cdot 10 = 30 \cdot 3290 - (\sum x)^2

STEP 6

Simplify the left side of the equation:
302910=303290(x)230 \cdot 29 \cdot 10 = 30 \cdot 3290 - (\sum x)^2
8700=98700(x)28700 = 98700 - (\sum x)^2

STEP 7

Rearrange the equation to solve for (x)2(\sum x)^2:
(x)2=987008700(\sum x)^2 = 98700 - 8700

STEP 8

Subtract to find (x)2(\sum x)^2:
(x)2=90000(\sum x)^2 = 90000

STEP 9

Take the square root of both sides to find x\sum x:
x=90000\sum x = \sqrt{90000}

STEP 10

Calculate the square root:
x=300\sum x = 300

STEP 11

Now that we have x\sum x, we can find the mean xˉ\bar{x} by dividing x\sum x by nn:
xˉ=xn\bar{x} = \frac{\sum x}{n}

STEP 12

Plug in the values for x\sum x and nn to calculate the mean:
xˉ=30030\bar{x} = \frac{300}{30}

STEP 13

Calculate the mean:
xˉ=10\bar{x} = 10
The mean of the observations is 10, which corresponds to option c.

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