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The following data gave \(X=\) the water content of snow on April 1 and \(Y=\) the yield from April to July (in inches) on the Snake River watershed in Wyoming for 1919 to 1935. (The data were taken from an article in Research Notes, Vol. \(61,1950,\) Pacific Northwest Forest Range Experiment Station, Oregon.) $$\begin{array}{cccc}\hline x & y & x & y \\\\\hline 23.1 & 10.5 & 37.9 & 22.8 \\\32.8 & 16.7 & 30.5 & 14.1 \\\31.8 & 18.2 & 25.1 & 12.9 \\ 32.0 & 17.0 & 12.4 & 8.8 \\\30.4 & 16.3 & 35.1 & 17.4 \\\24.0 & 10.5 & 31.5 & 14.9 \\\39.5 & 23.1 & 21.1 & 10.5 \\\24.2 & 12.4 & 27.6 & 16.1 \\\52.5 & 24.9 & & \\\\\hline\end{array}$$ (a) Estimate the correlation between \(Y\) and \(X\). (b) Test the hypothesis that \(\rho=0,\) using \(\alpha=0.05\). (c) Fit a simple linear regression model and test for significance of regression using \(\alpha=0.05 .\) What conclusions can you draw? How is the test for significance of regression related to the test on \(\rho\) in part (b)? (d) Analyze the residuals and comment on model adequacy.

Step by step solution

01

Calculate Correlation Coefficient (r)

First, we need to calculate the correlation coefficient to estimate the correlation between \( X \) and \( Y \) using the formula: \[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}} \]where \( n \) is the number of data pairs, \( \sum xy \) is the sum of the products of each pair, \( \sum x \) and \( \sum y \) are sums of \( x \) and \( y \).

02

Test Hypothesis for Correlation (蟻 = 0)

To test \( \rho = 0 \), use a t-test with the formula \[ t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}} \]and a two-tailed test with \( \alpha = 0.05 \) and \( n - 2 \) degrees of freedom. Compare the calculated t-value with the critical t-value from t-distribution tables.

03

Fit Simple Linear Regression Model

To fit the model \( Y = a + bX \), calculate:\[ b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \]\[ a = \frac{(\sum y) - b(\sum x)}{n} \]Use these to create the regression equation.

04

Test Significance of Regression

For the significance test, use the F-test by calculating \[ F = \frac{SSR/1}{SSE/(n-2)} \]where SSR is the regression sum of squares and SSE is the error sum of squares. Compare the F-value with a critical value from F-distribution tables with 1 and \( n-2 \) degrees of freedom.

05

Analyze Residuals

Calculate residuals (\( e_i = y_i - \hat{y_i} \)) and plot them to check for any patterns. Assess if residuals are normally distributed and independent with constant variance to evaluate model adequacy.

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Key Concepts

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

Linear Regression

Linear regression is a powerful statistical method used to model the relationship between a dependent variable and one or more independent variables. In this case, we are interested in the relation between the water content of snow (independent variable \(X\)) and the water yield (dependent variable \(Y\)).
To fit a linear regression model, we need to calculate the slope \(b\) and the intercept \(a\). These can be found using the formulas:

• \( b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \)
• \( a = \frac{(\sum y) - b(\sum x)}{n} \)

Once we have \(a\) and \(b\), we can form the regression equation: \(Y = a + bX\). This equation helps us predict the yield \(Y\) for a given water content \(X\).
In linear regression, the primary goal is understanding the strength and direction of this relationship, which can be expressed through the slope \(b\).
Moreover, the regression line minimizes the sum of squared differences between observed and predicted values, ensuring the best fit to the data according to the least squares criterion.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is significant evidence to reject a null hypothesis. In this exercise, we want to test if the correlation between water content and yield is zero (\( \rho = 0 \)). This null hypothesis assumes no linear relationship between \(X\) and \(Y\).
To perform this, we use a t-test for correlation. The test statistic is calculated as:

• \( t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}} \)

Here, \(r\) is the correlation coefficient calculated from the data, and \(n\) is the number of data pairs.
We compare the t-value to a critical value from t-distribution tables at a 0.05 significance level and \(n-2\) degrees of freedom. If the calculated t-value exceeds the critical value, we reject the null hypothesis, indicating a significant correlation.
This test ensures whether any observed relationship could still be due to random sampling variation rather than a true association.

Residual Analysis

Residual analysis involves studying the residuals, which are the differences between observed values and those predicted by the regression model. Residuals \(e_i\) are calculated as \(e_i = y_i - \hat{y_i}\), where \(y_i\) are the observed values and \(\hat{y_i}\) are the predicted values.
Through residual analysis, we check the assumptions of the linear regression model. Two key aspects analyzed in the residuals are:

• Whether the residuals are normally distributed.
• If there are patterns indicating non-linearity, suggesting that a simple linear model might not fit well.

Plotting residuals helps in visually inspecting these qualities. Ideally, residuals should display random scattering without any discernible patterns, showing constant variance across different levels of \(X\).
Residual analysis is crucial as it gives insight into the model's adequacy, revealing any violation of assumptions like homoscedasticity or normality.