91Ó°ÊÓ

Occasionally an investigator may wish to compute a confidence interval for \(\alpha\), the \(y\) intercept of the true regression line, or test hypotheses about \(\alpha .\) The estimated \(y\) intercept is simply the height of the estimated line when \(x=0\), since \(a+b(0)=a .\) This implies that \(s_{a}\) the estimated standard deviation of the statistic \(a\), results from substituting \(x^{\prime \prime}=0\) in the formula for \(s_{a+b x^{+}} .\) The desired confidence interval is then \(a \pm(t\) critical value \() s_{a}\) \(-\) and a test statistic is 1 $$ t=\frac{a-\text { hypothesized value }}{s_{a}} $$ a. The article "Comparison of Winter-Nocturnal Geostationary Satellite Infrared-Surface Temperature with Shelter-Height Temperature in Florida" (Remote Sensing of the Emvironment \([1983]: 313-327\) ) used the simple linear regression model to relate surface temperature as measured by a satellite \((y)\) to actual air temperature \((x)\) as determined from a thermocouple placed on a traversing vehicle. Selected data are given (read from a scatterplot in the article). \(\begin{array}{rrrrrrrr}x & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\ y & -3.9 & -2.1 & -2.0 & -1.2 & 0.0 & 1.9 & 0.6 \\ x & 5 & 6 & 7 & & & & \\ y & 2.1 & 1.2 & 3.0 & & & & \end{array}\) Estimate the true regression line. b. Compute the estimated standard deviation \(s_{a}\). Carry out a test at level of significance \(.05\) to see whether the \(y\) intercept of the true regression line differs from zero. c. Compute a \(95 \%\) confidence interval for \(\alpha\). Does the result indicate that \(\alpha=0\) is plausible? Explain.

Step by step solution

01

Estimate the true regression line

To get the estimated regression line, one has to calculate the slope and intercept of the line. First, calculate the mean of x values and y values. Then, calculate the slope (b) of the line using the formula \( b = \frac{\Sigma((x_i - \bar{x})(y_i - \bar{y}))}{\Sigma((x_i - \bar{x})^2)} \). Afterwards, calculate the intercept(a) using the formula \( a = \bar{y} - b\bar{x} \) where \bar{x} and \bar{y} indicate the mean of x and y values, respectively.

02

Calculate the estimated standard deviation \( s_a \)

The estimated standard deviation, \( s_a \), can be calculated using the formula \( s_a = \sqrt{\frac{\Sigma{(y_i - \hat{y_i})^2}}{n-2}} \), where \( n \) is the number of points, \( y_i \) are the actual y values, and \( \hat{y_i} \) are the predicted y values obtained from the model.

03

Conduct a hypothesis test

A hypothesis test is conducted to see if the y intercept \( \alpha \) could reasonably be 0. Here, null hypothesis is \( H_0: \alpha = 0 \). Alternative hypothesis is \( H_1: \alpha \neq 0 \). Then calculate the test statistic \( t = \frac{a - 0}{s_a} \), where 'a' is obtained from the regression line and \( s_a \) is the estimated standard deviation. The p-value associated with this test statistic is then calculated. The null hypothesis is rejected and it is concluded that \( \alpha \) is significantly different from 0, if the p-value is less than 0.05.

04

Calculate a 95% confidence interval for \( \alpha \)

A 95% confidence interval for \( \alpha \) can be calculated using the formula \( a \pm t_{0.025, n-2}s_a \), where 'a' is the y-intercept obtained from the regression line, and \( t_{0.025, n-2} \) is the critical t value for \( \alpha = 0.025 \) and degrees of freedom \( n-2 \). Validate the possibility of \( \alpha = 0 \) based on the range of the confidence interval. If the interval contains 0, then \( \alpha = 0 \) is plausible.

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Key Concepts

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

Confidence Interval

Understanding the concept of the confidence interval is crucial for interpreting the results from regression analysis accurately. A confidence interval gives us a range of values, usually calculated from the sample data, for which a parameter (like the y-intercept in our example) is expected to lie within a specific probability. For instance, a 95% confidence interval suggests that if we repeated the study multiple times, we would expect the true parameter to fall within this range 95% of the time.

Constructing a confidence interval involves determining the mean of the estimator and then calculating the margin of error, which hinges on the critical value from the t-distribution (linked to the confidence level and degrees of freedom in the sample data) and the standard deviation of the estimator. In our exercise, we calculate the 95% confidence interval for the y-intercept, \( \alpha \) of the true regression line, giving us insights into the plausible values that \( \alpha \) can take, considering the random variation present in the sample data.

Hypothesis Testing

Hypothesis testing in statistics is a method used to make decisions or inferences about a population parameter based on sample data. In the context of regression analysis, we may want to test hypotheses about the y-intercept, \( \alpha \). For example, we may hypothesize that the y-intercept is equal to zero (no effect or relationship), which is our null hypothesis (\( H_0: \alpha = 0 \)).

The alternative hypothesis (\( H_1 \)) would state that the y-intercept is not equal to zero (signifying a possible effect or relationship). We use a test statistic, like the t-statistic, to determine whether to reject the null hypothesis. This test statistic takes into account both the magnitude of the difference from the hypothesized value and the variability of the estimate, expressed through the standard deviation. Results with a p-value less than the significance level, such as 0.05, suggest rejecting the null hypothesis, indicating that the y-intercept is significantly different from the hypothesized value.

Standard Deviation

Standard deviation is a measure that quantifies the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value), whereas a high standard deviation indicates that the values are spread out over a wider range.

In regression analysis, the standard deviation of the y-intercept, \( s_{\alpha} \), reflects the extent to which the intercepts obtained from different sample data would spread around the true y-intercept of the population regression line. The smaller the standard deviation, the more precise our estimate is likely to be. We use this measure of spread to compute both the margin of error for confidence intervals and the test statistic for hypothesis testing, as seen in the given exercise.

Y-Intercept

In the context of a linear regression model, the y-intercept (often denoted as \( \alpha \) or 'a') is a critical parameter representing the value of the dependent variable when all independent variables are equal to zero. It is the starting point of the regression line when it crosses the y-axis on a graph.

If we're analyzing the relationship between surface temperatures from satellite data (y) and actual air temperatures (x), the y-intercept will give us an estimate of what the satellite temperature reading would be when the actual air temperature is zero. Estimating the y-intercept with precision is vital, as it sets the baseline from which we assess the effect of the independent variable(s) on the dependent variable. As shown in our exercise, by using the mean values of x and y and the slope of the regression, we can solve for the estimated y-intercept of the model.