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A consumer buying cooperative tested the effective heating area of 20 different electric space heaters with different wattages. Here are the results. $$ \begin{array}{|crr|rrr|} \hline \text { Heater } & \text { Wattage } & \text { Area } & \text { Heater } & \text { Wattage } & \text { Area } \\ \hline 1 & 1,500 & 205 & 11 & 1,250 & 116 \\ 2 & 750 & 70 & 12 & 500 & 72 \\ 3 & 1,500 & 199 & 13 & 500 & 82 \\ 4 & 1,250 & 151 & 14 & 1,500 & 206 \\ 5 & 1,250 & 181 & 15 & 2,000 & 245 \\ 6 & 1,250 & 217 & 16 & 1,500 & 219 \\ 7 & 1,000 & 94 & 17 & 750 & 63 \\ 8 & 2,000 & 298 & 18 & 1,500 & 200 \\ 9 & 1,000 & 135 & 19 & 1,250 & 151 \\ 10 & 1,500 & 211 & 20 & 500 & 44 \\ \hline \end{array} $$ a. Compute the correlation between the wattage and heating area. Is there a direct or an indirect relationship? b. Conduct a test of hypothesis to determine if it is reasonable that the coefficient is greater than zero. Use the .05 significance level. c. Develop the regression equation for effective heating based on wattage. d. Which heater looks like the "best buy" based on the size of the residual?

Step by step solution

01

Organize Data for Correlation

List the wattage and area pairs from the data. For instance, (1500, 205), (750, 70), etc., until (500, 44). These will be used to calculate the correlation coefficient.

02

Calculate the Correlation Coefficient

Use the formula for Pearson's correlation coefficient, \( r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}\)where \(n\) is the number of heaters, \(x\) is wattage, \(y\) is area. After computing values, the correlation coefficient \(r\) will tell us the strength and direction of the relationship between wattage and area.

03

Interpret Correlation Result

Evaluate the value of \(r\) calculated in the previous step. If \(r > 0\), there is a direct relationship; if \(r < 0\), there is an indirect relationship. Usually, \(r\) values close to 1 or -1 imply a strong relationship, while values near 0 imply a weak relationship.

04

Formulate Hypothesis for Significance Test

State the null hypothesis \(H_0: \rho = 0\) and the alternative hypothesis \(H_1: \rho > 0\). Here, \(\rho\) represents the population correlation coefficient. We want to determine if there's sufficient evidence to conclude \(\rho\) is greater than zero.

05

Calculate Test Statistic

Utilize the formula for the test statistic,\[t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}\]Determine the degrees of freedom as \(n-2\), which is 18 for this data. Compute the \(t\)-value using the previously found \(r\) to assess its significance.

06

Compare Test Statistic to Significance Level

Compare the computed \(t\)-value with the critical \(t\)-value at 0.05 significance level from the \(t\)-distribution table with 18 degrees of freedom. If the \(t\)-value is greater than the critical value, we reject \(H_0\).

07

Develop Regression Equation

Use the least squares method to derive the regression line equation, \(y = mx + b\), where \(m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}\) and \(b = \frac{\sum y - m\sum x}{n}\).Insert the sums and \(n\) to solve for \(m\) and \(b\) to get the equation.

08

Identify Best Buy Using Residuals

Calculate residuals for each heater using \(| ext{actual area} - ext{predicted area}|\), and identify the heater with the smallest residual. This heater is the best buy since its actual heating area is closest to the predicted value.

09

Conclusion

Summarize results: The correlation indicates a (possibly strong) direct relationship; the hypothesis test supports this relationship as significant; the regression equation allows prediction of heating area from wattage, and the best buy is determined by minimal residual.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a statistical method that allows us to determine if there is enough evidence to support a given claim about a population parameter. In our exercise involving electric heaters, we want to test the hypothesis about the correlation between wattage and heating area.

To begin, we set up two hypotheses:

• Null hypothesis (\(H_0\)): There is no real correlation between wattage and heating area, or \(\rho = 0\).
• Alternative hypothesis (\(H_1\)): There is a positive correlation, meaning \(\rho > 0\).

We use a significance level of \(0.05\), which indicates a 5% risk of concluding that there is a correlation when there is none. The next step in hypothesis testing is to use the calculated test statistic, which, in this case, involves Pearsons's \(r\) coefficient, to determine whether the evidence supports rejecting the null hypothesis. A test statistic greater than the critical value from the \(t\)-distribution implies that the null hypothesis should be rejected, indicating a significant positive correlation.

Pearson's Correlation Coefficient

Pearson's correlation coefficient, denoted by \(r\), is a measure of the linear relationship between two variables. In this study, we analyze the relationship between the wattage of space heaters and their effective heating area.

The formula for Pearson's \(r\) is given by:\[r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}\]where:

• \(x\) is the wattage
• \(y\) is the heating area
• \(n\) is the sample size

The value of \(r\) ranges from \(-1\) to \(1\) with:

• \(r = 1\) indicating a perfect positive linear relationship
• \(r = -1\) indicating a perfect negative linear relationship
• \(r = 0\) indicating no linear relationship

In this exercise, calculating \(r\) helps us understand if wattage increases, does the heating area also increase significantly.

Regression Equation

Creating a regression equation involves finding the best-fit line through the data that allows us to predict the outcome variable, known as the dependent variable, from the independent variable.

In our exercise, the regression equation aims to predict heating area (\(y\)) based on wattage (\(x\)). The equation for the line is:\[ y = mx + b \]Here,

• \(m\) represents the slope, calculated as:
• \[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}\]
• \(b\) is the y-intercept, calculated as:
• \[ b = \frac{\sum y - m\sum x}{n} \]

By inserting the sums and the number of heaters (\(n\)) into these formulas, you can solve for \(m\) and \(b\) to get the specific equation. This allows us to forecast the expected heating area for any given wattage, assuming the relationship holds.

Residuals and Best Fit

Residuals are crucial in evaluating how well a regression line fits the data. A residual is the difference between the observed value and the value predicted by the regression line.

To compute residuals:

• Calculate the predicted heating area using the regression equation for each heater.
• Compute the residual as \(|\text{actual area} - \text{predicted area}|\)

Analyzing residuals allows us to see how accurately the regression line represents the data.

For our exercise, the heater with the smallest residual is considered the "best buy." This means its actual performance is closest to what the regression model predicts, implying reliability and consistency in effective heating comparable to other models.