Solved on Feb 17, 2024

Compute the correlation coefficient $r$ given $\sum x=5,811, \sum x^{2}=5,632,643, \sum y=578, \sum y^{2}=65,292$ and $\sum x y=553,170$ . Does $r$ imply that $y$ should tend to increase, decrease, or remain constant as $x$ increases?

STEP 1

Assumptions
1. We have the following sums for a set of data points: $\sum x=5,811$ , $\sum x^{2}=5,632,643$ , $\sum y=578$ , $\sum y^{2}=65,292$ , and $\sum x y=553,170$ .
2. The number of data points, $n$ , is not given and needs to be calculated or assumed to be known.
3. The correlation coefficient, $r$ , is calculated using the formula: $r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}}$
4. The correlation coefficient, $r$ , indicates the strength and direction of a linear relationship between two variables.

STEP 2

First, we need to find the number of data points, $n$ . Since it is not provided, we will assume it is known or calculate it based on the information given.

STEP 3

We will use the formula for the correlation coefficient, $r$ , to compute its value. We will plug in the sums provided into the formula.

STEP 4

Calculate the numerator of the correlation coefficient formula:
$numerator = n\sum xy - \sum x \sum y$

STEP 5

Using the provided sums, the numerator becomes:
$numerator = n \cdot 553,170 - 5,811 \cdot 578$

STEP 6

Calculate the denominator of the correlation coefficient formula:
$denominator = \sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}$

STEP 7

Using the provided sums, the denominator becomes:
$denominator = \sqrt{(n \cdot 5,632,643 - 5,811^2)(n \cdot 65,292 - 578^2)}$

STEP 8

Now, we need to find the value of $n$ . If $n$ is not provided, we must assume it or deduce it from the context of the problem. Since the problem doesn't give us any information about $n$ , we will assume it is known for the sake of calculating $r$ .

STEP 9

Assuming $n$ is known, we can now calculate the numerator and denominator with the actual value of $n$ .

STEP 10

Once we have the numerator and denominator, we can calculate $r$ by dividing the numerator by the denominator:
$r = \frac{numerator}{denominator}$

STEP 11

After calculating $r$ , round the answer to four decimal places as requested.

STEP 12

To determine whether $y$ tends to increase or decrease as $x$ increases, we look at the sign of $r$ . If $r$ is positive, it implies that as $x$ increases, $y$ tends to increase. If $r$ is negative, it implies that as $x$ increases, $y$ tends to decrease. If $r$ is close to zero, it implies that there is no strong linear relationship between $x$ and $y$ .

STEP 13

Based on the calculated value of $r$ , we will determine the tendency of $y$ as $x$ increases.

STEP 14

Since we do not have the actual value of $n$ , we cannot compute the exact value of $r$ and thus cannot conclude the behavior of $y$ as $x$ increases. However, if we had $n$ , we would follow the steps above to compute $r$ and make the conclusion.

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Compute the correlation coefficient $r$ given $\sum x=5,811, \sum x^{2}=5,632,643, \sum y=578, \sum y^{2}=65,292$ and $\sum x y=553,170$ . Does $r$ imply that $y$ should tend to increase, decrease, or remain constant as $x$ increases?

STEP 1

STEP 2

First, we need to find the number of data points, $n$ . Since it is not provided, we will assume it is known or calculate it based on the information given.

STEP 3

We will use the formula for the correlation coefficient, $r$ , to compute its value. We will plug in the sums provided into the formula.

STEP 4

Calculate the numerator of the correlation coefficient formula:
$numerator = n\sum xy - \sum x \sum y$

STEP 5

Using the provided sums, the numerator becomes:
$numerator = n \cdot 553,170 - 5,811 \cdot 578$

STEP 6

Calculate the denominator of the correlation coefficient formula:
$denominator = \sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}$

STEP 7

Using the provided sums, the denominator becomes:
$denominator = \sqrt{(n \cdot 5,632,643 - 5,811^2)(n \cdot 65,292 - 578^2)}$

STEP 8

Now, we need to find the value of $n$ . If $n$ is not provided, we must assume it or deduce it from the context of the problem. Since the problem doesn't give us any information about $n$ , we will assume it is known for the sake of calculating $r$ .

STEP 9

Assuming $n$ is known, we can now calculate the numerator and denominator with the actual value of $n$ .

STEP 10

Once we have the numerator and denominator, we can calculate $r$ by dividing the numerator by the denominator:
$r = \frac{numerator}{denominator}$

STEP 11

After calculating $r$ , round the answer to four decimal places as requested.

STEP 12

STEP 13

Based on the calculated value of $r$ , we will determine the tendency of $y$ as $x$ increases.

STEP 14

Since we do not have the actual value of $n$ , we cannot compute the exact value of $r$ and thus cannot conclude the behavior of $y$ as $x$ increases. However, if we had $n$ , we would follow the steps above to compute $r$ and make the conclusion.

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STEP 1

STEP 2

First, we need to find the number of data points, nnn. Since it is not provided, we will assume it is known or calculate it based on the information given.

STEP 3

We will use the formula for the correlation coefficient, rrr, to compute its value. We will plug in the sums provided into the formula.

STEP 4

Calculate the numerator of the correlation coefficient formula:numerator=n∑xy−∑x∑y numerator = n\sum xy - \sum x \sum y numerator=n∑xy−∑x∑y

STEP 5

Using the provided sums, the numerator becomes:numerator=n⋅553,170−5,811⋅578 numerator = n \cdot 553,170 - 5,811 \cdot 578 numerator=n⋅553,170−5,811⋅578

STEP 6

Calculate the denominator of the correlation coefficient formula:denominator=(n∑x2−(∑x)2)(n∑y2−(∑y)2) denominator = \sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)} denominator=(n∑x2−(∑x)2)(n∑y2−(∑y)2)​

STEP 7

Using the provided sums, the denominator becomes:denominator=(n⋅5,632,643−5,8112)(n⋅65,292−5782) denominator = \sqrt{(n \cdot 5,632,643 - 5,811^2)(n \cdot 65,292 - 578^2)} denominator=(n⋅5,632,643−5,8112)(n⋅65,292−5782)​

STEP 8

Now, we need to find the value of nnn. If nnn is not provided, we must assume it or deduce it from the context of the problem. Since the problem doesn't give us any information about nnn, we will assume it is known for the sake of calculating rrr.

STEP 9

Assuming nnn is known, we can now calculate the numerator and denominator with the actual value of nnn.

STEP 10

Once we have the numerator and denominator, we can calculate rrr by dividing the numerator by the denominator:r=numeratordenominator r = \frac{numerator}{denominator} r=denominatornumerator​

STEP 11

After calculating rrr, round the answer to four decimal places as requested.

STEP 12

STEP 13

Based on the calculated value of rrr, we will determine the tendency of yyy as xxx increases.

STEP 14

Since we do not have the actual value of nnn, we cannot compute the exact value of rrr and thus cannot conclude the behavior of yyy as xxx increases. However, if we had nnn, we would follow the steps above to compute rrr and make the conclusion.

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First, we need to find the number of data points, $n$ . Since it is not provided, we will assume it is known or calculate it based on the information given.

We will use the formula for the correlation coefficient, $r$ , to compute its value. We will plug in the sums provided into the formula.

Calculate the numerator of the correlation coefficient formula:
$numerator = n\sum xy - \sum x \sum y$

Using the provided sums, the numerator becomes:
$numerator = n \cdot 553,170 - 5,811 \cdot 578$

Calculate the denominator of the correlation coefficient formula:
$denominator = \sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}$

Using the provided sums, the denominator becomes:
$denominator = \sqrt{(n \cdot 5,632,643 - 5,811^2)(n \cdot 65,292 - 578^2)}$

Now, we need to find the value of $n$ . If $n$ is not provided, we must assume it or deduce it from the context of the problem. Since the problem doesn't give us any information about $n$ , we will assume it is known for the sake of calculating $r$ .

Assuming $n$ is known, we can now calculate the numerator and denominator with the actual value of $n$ .

Once we have the numerator and denominator, we can calculate $r$ by dividing the numerator by the denominator:
$r = \frac{numerator}{denominator}$

After calculating $r$ , round the answer to four decimal places as requested.

Based on the calculated value of $r$ , we will determine the tendency of $y$ as $x$ increases.

Since we do not have the actual value of $n$ , we cannot compute the exact value of $r$ and thus cannot conclude the behavior of $y$ as $x$ increases. However, if we had $n$ , we would follow the steps above to compute $r$ and make the conclusion.