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The catch basin in a storm-sewer system is the interface between surface runoff and the sewer. The catch-basin insert is a device for retrofitting catch basins to improve pollutantremoval properties. The article "An Evaluation of the Urban Stormwater Pollutant Removal Efficiency of Catch Basin Inserts" (Water Envir: Res., 2005: 500-510) reported on tests of various inserts under controlled conditions for which inflow is close to what can be expected in the field. Consider the following data, read from a graph in the article, for one particular type of insert on \(x=\) amount filtered ( 1000 s of liters) and \(y=\%\) total suspended solids removed. $$ \begin{array}{l|cccccccccc} x & 23 & 45 & 68 & 91 & 114 & 136 & 159 & 182 & 205 & 228 \\ \hline y & 53.3 & 26.9 & 54.8 & 33.8 & 29.9 & 8.2 & 17.2 & 12.2 & 3.2 & 11.1 \end{array} $$ Summary quantities are $$ \begin{aligned} &\sum x_{i}=1251, \sum x_{i}^{2}=199,365, \sum y_{i}=250.6 \\ &\sum y_{i}^{2}=9249.36, \sum x_{i} y_{i}=21,904.4 \end{aligned} $$ a. Does a scatter plot support the choice of the simple linear regression model? Explain. b. Obtain the equation of the least squares line. c. What proportion of observed variation in \% removed can be attributed to the model relationship? d. Does the simple linear regression model specify a useful relationship? Carry out an appropriate test of hypotheses using a significance level of \(.05\). e. Is there strong evidence for concluding that there is at least a \(2 \%\) decrease in true average suspended solid removal associated with a 10,000 liter increase in the amount filtered? Test appropriate hypotheses using \(\alpha=.05\). f. Calculate and interpret a \(95 \% \mathrm{CI}\) for true average \(\%\) removed when amount filtered is 100,000 liters. How does this interval compare in width to a CI when amount filtered is 200,000 liters? g. Calculate and interpret a \(95 \%\) PI for \% removed when amount filtered is 100,000 liters. How does this interval compare in width to the CI calculated in (f) and to a PI when amount filtered is 200,000 liters?

Step by step solution

01

Evaluate the Scatter Plot Support for Linear Regression

Using the given data points of x and y, construct a scatter plot. Examine the data distribution to determine if a linear pattern or trend appears. A linear trend would suggest the appropriateness of a simple linear regression model.

02

Calculate the Least Squares Line

Use the given sums to calculate the slope (b) and y-intercept (a) of the least squares line using the formulas: \( b = \frac{\sum xy - \frac{\sum x \sum y}{n}}{\sum x^2 - \frac{(\sum x)^2}{n}} \) and \( a = \frac{\sum y - b \sum x}{n} \). Apply these calculations to find the best-fit line equation.

03

Determine Proportion of Variation Explained

Calculate the coefficient of determination \(R^2\) using the formula: \( R^2 = \frac{[\sum xy - \frac{\sum x \sum y}{n}]^2}{ [\sum x^2 - \frac{(\sum x)^2}{n}][\sum y^2 - \frac{(\sum y)^2}{n}]} \). \(R^2\) represents the proportion of variability in the percentage of suspended solids removed explained by the linear model.

04

Hypothesis Testing for Model Usefulness

Conduct a hypothesis test for \( H_0: b = 0 \) versus \( H_a: b eq 0 \) using a significance level of 0.05. Calculate the test statistic and compare it to the critical value or use the p-value approach to determine if the model is statistically significant.

05

Test for Specific Decrease in Removal Efficiency

Formulate hypotheses for a decrease of at least 2% removal efficiency for each 10,000-liter increase: \( H_0: b \geq -0.20 \) versus \( H_a: b < -0.20 \). Conduct a one-tailed t-test using a significance level of 0.05, and interpret the results.

06

Confidence Interval for True Average at 100,000 Liters

Calculate a 95% confidence interval for the true average percentage removed when the amount filtered is 100,000 liters. Use the formula for CI involving the predicted value, standard error, and t-distribution.

07

Prediction Interval for Percentage Removed at 100,000 Liters

Calculate a 95% prediction interval for the percentage removed when the amount filtered is 100,000 liters. Compare this interval's width with the confidence interval from step 6 and evaluate any differences.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

scatter plot

A scatter plot is a type of graph used to represent individual data points in two variables, usually shown as x and y coordinates on a Cartesian plane. It helps visualize the relationship between these two variables. By plotting the given data points, you can identify patterns or trends, which indicate if a simple linear regression model is suitable.

• A linear pattern in the scatter plot suggests that a simple linear regression model can be used to describe the relationship between the variables.
• If the points roughly follow a straight line, either upward or downward, this indicates a linear trend.
• However, if the points do not display any clear linear arrangement, a simple linear regression model might not be appropriate.

In the context of the exercise, plotting the percentage of solids removed against the amount filtered will help show you if a linear relationship exists.

least squares line

The least squares line is a statistical method used to determine the line of best fit for a given set of data points. Essentially, it minimizes the sum of the squares of the vertical distances (residuals) between each data point and the line itself. This process ensures the line is as close as possible to the data.

• The slope (b) of the line indicates how much y is expected to change when x increases by one unit.
• The y-intercept (a) is the predicted value of y when x is zero.
• Use the least squares formulas: \[ b = \frac{\sum xy - \frac{\sum x \sum y}{n}}{\sum x^2 - \frac{(\sum x)^2}{n}} \] and \[ a = \frac{\sum y - b \sum x}{n} \] to calculate these parameters.

This line is central in predicting outcomes and understanding relationships in the data.

coefficient of determination

The coefficient of determination, often represented as \( R^2 \), is a measure that assesses how well a model explains and predicts future outcomes. It's a key indicator in regression analysis.

• \( R^2 \) ranges from 0 to 1, where 0 means the model explains none of the variability of the response data around its mean and 1 means it explains all the variability.
• An \( R^2 \) value closer to 1 implies a good fit between the model and the observed data.
• Calculate \( R^2 \) using \[ R^2 = \frac{[\sum xy - \frac{\sum x \sum y}{n}]^2}{[\sum x^2 - \frac{(\sum x)^2}{n}][\sum y^2 - \frac{(\sum y)^2}{n}]} \]

In simple linear regression, \( R^2 \) tells us the proportion of variance in the dependent variable that can be explained by the independent variable.

hypothesis testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. In the context of regression, hypothesis testing can be used to determine if there is a significant relationship between the independent and dependent variables.

• In regression, a common test checks the null hypothesis \( H_0 \): that the slope (b) equals zero (no relationship), against the alternative hypothesis \( H_a \): that b is not zero (there is a relationship).
• When using a significance level of 0.05, it indicates a 5% risk of concluding a relationship exists when there is none.
• If the p-value obtained is less than 0.05, the null hypothesis is rejected, suggesting the regression model is useful.

This testing helps establish the validity and reliability of the model used in the exercise.

confidence interval

A confidence interval gives a range of values, derived from sample data, within which a population parameter is expected to lie. It is expressed with a certain level of confidence.

• A 95% confidence interval means that if the same population is sampled multiple times, 95% of the intervals computed from that sample data will contain the true parameter value.
• In regression, confidence intervals can be calculated for the average predicted values of y for a given x.
• The width of a confidence interval depends on the variability of the data, the sample size, and the confidence level chosen.

For this exercise, you compare the CI widths when predicting the percentage of solids removed for different amounts of filtered water, showing how data variability affects prediction accuracy.