Solved on Feb 19, 2024

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

STEP 1

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

STEP 2

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

STEP 3

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

STEP 4

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

STEP 5

Plug in the given values for $S^2$ , $n$ , and $\sum x^2$ into the equation:
$30(30-1) \cdot 10 = 30 \cdot 3290 - (\sum x)^2$

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

STEP 11

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

STEP 12

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

STEP 13

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

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

STEP 1

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

STEP 2

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

STEP 3

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

STEP 4

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

STEP 5

Plug in the given values for $S^2$ , $n$ , and $\sum x^2$ into the equation:
$30(30-1) \cdot 10 = 30 \cdot 3290 - (\sum x)^2$

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

STEP 11

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

STEP 12

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

STEP 13

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

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

STEP 1

Assumptions1. The sample variance S2S^2S2 is 10.2. The sample size nnn is 30.3. The sum of the squares of the observations ∑x2\sum x^2∑x2 is 3290.4. We are asked to find the mean of the observations.

STEP 2

Recall the formula for the sample variance S2S^2S2:S2=1n−1(∑x2−(∑x)2n)S^2 = \frac{1}{n-1} \left(\sum x^2 - \frac{(\sum x)^2}{n}\right)S2=n−11​(∑x2−n(∑x)2​)

STEP 3

STEP 4

Multiply both sides by nnn to clear the fraction:n(n−1)S2=n∑x2−(∑x)2n(n-1)S^2 = n\sum x^2 - (\sum x)^2n(n−1)S2=n∑x2−(∑x)2

STEP 5

Plug in the given values for S2S^2S2, nnn, and ∑x2\sum x^2∑x2 into the equation:30(30−1)⋅10=30⋅3290−(∑x)230(30-1) \cdot 10 = 30 \cdot 3290 - (\sum x)^230(30−1)⋅10=30⋅3290−(∑x)2

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

STEP 11

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

STEP 12

Plug in the values for ∑x\sum x∑x and nnn to calculate the mean:xˉ=30030\bar{x} = \frac{300}{30}xˉ=30300​

STEP 13

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

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

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

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

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

Plug in the given values for $S^2$ , $n$ , and $\sum x^2$ into the equation:
$30(30-1) \cdot 10 = 30 \cdot 3290 - (\sum x)^2$

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

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

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

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

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

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

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

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