Some types of algae have the potential to cause damage to river ecosystems. The accompanying data on y = algae colony density and x = rock surface area for nine rivers are a subset of data that come from a study.

x	51	55	49	79	43	37	69	45	48

y	154	48	24	35	36	171	11	185	27

What is the value of 

se

 for this data set? (Round your answer to five decimal places.)

To calculate the value of \( s_e \), which is the standard error of the estimate, we first need to compute the regression equation (i.e., the line of best fit) for the data. The standard error \( s_e \) is given by:

\[
s_e = \sqrt{\frac{ \sum (y_i - \hat{y}_i)^2 }{n-2}}
\]

Where:
- \( y_i \) are the observed values,
- \( \hat{y}_i \) are the predicted values from the regression line,
- \( n \) is the number of data points.

### Steps:
1. Calculate the regression line equation \( y = a + bx \), where \( b \) is the slope and \( a \) is the intercept.
2. Use the regression line to compute the predicted values \( \hat{y}_i \).
3. Calculate the residuals \( y_i - \hat{y}_i \) for each data point.
4. Compute the sum of squared residuals \( \sum (y_i - \hat{y}_i)^2 \).
5. Finally, compute \( s_e \) using the formula above.

I'll calculate that for you.The standard error of the estimate (\( s_e \)) for the given data set is approximately 65.20796 (rounded to five decimal places).

Would you like further details or have any questions about the calculation?

Here are five related questions to expand your understanding:
1. How is the slope of the regression line calculated in linear regression?
2. What does the residual value represent in regression analysis?
3. Why is \( n - 2 \) used in the denominator when calculating the standard error of the estimate?
4. How can you interpret the standard error of the estimate in terms of model fit?
5. What is the relationship between the correlation coefficient and the goodness of fit?

**Tip:** The smaller the standard error of the estimate, the better the regression line fits the data.

Some types of algae have the potential to cause damage to river ecosystems. The accompanying data on y = algae colony density and x = rock surface area for nine rivers are a subset of data that come from a study.

x: 51, 55, 49, 79, 43, 37, 69, 45, 48

y: 154, 48, 24, 35, 36, 171, 11, 185, 27

What is the value of se for this data set? (Round your answer to five decimal places.)

Calculate the standard error of estimate for a dataset examining the relationship between algae colony density and rock surface area in nine rivers. Learn how to apply regression analysis and compute the residuals and standard error. Suitable for college-level statistics students.

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