Math / Data & Statistics

Question\begin{problem} The governor of a state has put together a team tasked with determining factors that account for the number of children living in poverty within the state. The team wants to know if the number of children living in poverty in a town is proportional to the population of the town, so they look at the population and number of children in poverty for 10 towns in the state. The data is reported in the table below.
\begin{center} \begin{tabular}{|c|c|} \hline Population & Children in Poverty \\ \hline 41,788 & 992 \\ 8,767 & 41 \\ 59,376 & 702 \\ 2,920 & 17 \\ 2,862 & 31 \\ 16,344 & 114 \\ 9,099 & 170 \\ 92,513 & 1,239 \\ 10,354 & 105 \\ 31,705 & 625 \\ \hline \end{tabular} \end{center}
\begin{enumerate} \item[(a)] What is the equation of the line of best fit? \item[(b)] What is " $r$ " and determine if it is a strong, moderate, or weak correlation. \end{enumerate} \end{problem}

Studdy Solution

STEP 1

1. The data consists of two variables: Population and Children in Poverty.
2. We are assuming a linear relationship between the two variables for the purpose of finding the line of best fit.
3. The correlation coefficient $r$ will be used to determine the strength of the linear relationship.

STEP 2

1. Calculate the line of best fit using the least squares method.
2. Calculate the correlation coefficient $r$ .
3. Interpret the value of $r$ to determine the strength of the correlation.

STEP 3

Calculate the means of the Population and Children in Poverty data sets.
Let $x_i$ represent the Population data and $y_i$ represent the Children in Poverty data.
Calculate the mean of $x$ :
$\bar{x} = \frac{41,788 + 8,767 + 59,376 + 2,920 + 2,862 + 16,344 + 9,099 + 92,513 + 10,354 + 31,705}{10}$
Calculate the mean of $y$ :
$\bar{y} = \frac{992 + 41 + 702 + 17 + 31 + 114 + 170 + 1,239 + 105 + 625}{10}$

STEP 4

Calculate the slope $m$ of the line of best fit using the formula:
$m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$

STEP 5

Calculate the y-intercept $b$ of the line of best fit using the formula:
$b = \bar{y} - m\bar{x}$

STEP 6

Calculate the correlation coefficient $r$ using the formula:
$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$

STEP 7

Interpret the value of $r$ :
- If $|r|$ is close to 1, the correlation is strong. - If $|r|$ is around 0.5, the correlation is moderate. - If $|r|$ is close to 0, the correlation is weak.
The equation of the line of best fit is $y = mx + b$ where $m$ and $b$ are calculated in Steps 2 and 3. The correlation coefficient $r$ is calculated in Step 4 and interpreted in Step 5.

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Studdy Solution

STEP 1

1. The data consists of two variables: Population and Children in Poverty.
2. We are assuming a linear relationship between the two variables for the purpose of finding the line of best fit.
3. The correlation coefficient $r$ will be used to determine the strength of the linear relationship.

STEP 2

1. Calculate the line of best fit using the least squares method.
2. Calculate the correlation coefficient $r$ .
3. Interpret the value of $r$ to determine the strength of the correlation.

STEP 3

STEP 4

Calculate the slope $m$ of the line of best fit using the formula:
$m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$

STEP 5

Calculate the y-intercept $b$ of the line of best fit using the formula:
$b = \bar{y} - m\bar{x}$

STEP 6

Calculate the correlation coefficient $r$ using the formula:
$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$

STEP 7

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Studdy Solution

STEP 1

1. The data consists of two variables: Population and Children in Poverty.2. We are assuming a linear relationship between the two variables for the purpose of finding the line of best fit.3. The correlation coefficient r r r will be used to determine the strength of the linear relationship.

STEP 2

1. Calculate the line of best fit using the least squares method.2. Calculate the correlation coefficient r r r.3. Interpret the value of r r r to determine the strength of the correlation.

STEP 3

STEP 4

Calculate the slope m m m of the line of best fit using the formula:m=∑(xi−xˉ)(yi−yˉ)∑(xi−xˉ)2m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}m=∑(xi​−xˉ)2∑(xi​−xˉ)(yi​−yˉ​)​

STEP 5

Calculate the y-intercept b b b of the line of best fit using the formula:b=yˉ−mxˉb = \bar{y} - m\bar{x}b=yˉ​−mxˉ

STEP 6

Calculate the correlation coefficient r r r using the formula:r=∑(xi−xˉ)(yi−yˉ)∑(xi−xˉ)2∑(yi−yˉ)2r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}r=∑(xi​−xˉ)2∑(yi​−yˉ​)2​∑(xi​−xˉ)(yi​−yˉ​)​

STEP 7

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1. The data consists of two variables: Population and Children in Poverty.
2. We are assuming a linear relationship between the two variables for the purpose of finding the line of best fit.
3. The correlation coefficient $r$ will be used to determine the strength of the linear relationship.

1. Calculate the line of best fit using the least squares method.
2. Calculate the correlation coefficient $r$ .
3. Interpret the value of $r$ to determine the strength of the correlation.

Calculate the slope $m$ of the line of best fit using the formula:
$m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$

Calculate the y-intercept $b$ of the line of best fit using the formula:
$b = \bar{y} - m\bar{x}$

Calculate the correlation coefficient $r$ using the formula:
$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$