Question

A sample of \(n=61\) penguin burrows was selected. and values of both \(y=\) trail length \((\mathrm{m})\) and \(x=\) soil hardness (force required to penetrate the substrate to a depth of \(12 \mathrm{~cm}\) with a certain gauge, in \(\mathrm{kg}\) ) were determined for each one ("Effects of Substrate on the Distribution of Magellanic Penguin Burrows," The Auk [1991]: 923-933). The equation of the least-squares line was \(\hat{y}=11.607-\) \(1.4187 x\), and \(r^{2}=.386\). a. Does the relationship between soil hardness and trail length appear to be linear, with shorter trails associated with harder soil (as the article asserted)? Carry out an appropriate test of hypotheses. b. Using \(s_{e}=2.35, \bar{x}=4.5\), and \(\sum(x-\bar{x})^{2}=250\), predict trail length when soil hardness is \(6.0\) in a way that conveys information about the reliability and precision of the prediction. c. Would you use the simple linear regression model to predict trail length when hardness is \(10.0 ?\) Explain your reasoning.

Short answer

a) Yes, the relationship between soil hardness and trail length appears to be linear and negative, although a p-value calculation is needed to confirm statistical significance. b) Inserting x=6.0 into the equation gives the predicted trail length and a prediction interval can illustrate the reliability and precision. c) It's not advisable to use the simple linear regression model to predict trail length for a soil hardness of 10.0 because it falls outside the range of the given data.

Step-by-step solution

- Understanding Linear Relationship

Firstly, a negative coefficient for x in the equation signifies that as x (soil hardness) increases, y (trail length) decreases, which indicates an inverse relationship. The p-value for this hypothesis test would be used to determine if this relationship is statistically significant. If the p-value is less than the significance level (commonly 0.05), then the relationship is considered statistically significant. To find the p-value, the correlation coefficient r is needed, which can be found by taking the square root of the given \(r^{2} = 0.386\). Then, the correlation coefficient is used with a t-distribution table or online calculator for a two-tailed test with degrees of freedom df=n-2 to find the p-value.

- Predicting Trail Length

The trail length when soil hardness is 6.0 can be found by inputting the given x-value into the regression equation: \(\hat{y} = 11.607 - 1.4187 * 6.0\). The result gives the predicted trail length. To indicate reliability and precision, we can also calculate a prediction interval. The formula for a prediction interval is: \(\hat{y} \pm t*se*\sqrt{1+1/n+(x-\bar{x})^2/}\), where t is the t-value corresponding to the desired confidence level from a t-distribution table with df=n-2, se is the standard error, n is the number of samples, x is the given x-value, and \(\bar{x}\) is the mean x-value.

- Assessing Applicability of the Regression Model

The simple linear regression model bases predictions on the assumption that the relationship between x and y is linear within the given data range. However, predicting the trail length for soil hardness of 10.0 may not be accurate because it's more than one standard deviation beyond the mean x-value in the data set. Additionally, extrapolation beyond the scope of the data used for model creation can often lead to incorrect predictions. Therefore, it's typically not recommended to use this model to make predictions for for soil hardness of 10.0.

Key concepts

Linear Relationship

Understanding the concept of a linear relationship is fundamental when dealing with simple linear regression. In the context of our exercise, a linear relationship between soil hardness (independent variable, denoted as 'x') and penguin trail length (dependent variable, denoted as 'y') is suggested by the equation \(\hat{y}=11.607-1.4187x\).

A linear relationship implies that a change in the independent variable will cause a proportional change in the dependent variable, with the relationship between the two variables being represented by a straight line when plotted on a graph. This line is known as the 'regression line' or 'least-squares line', which minimizes the sum of squared differences between observed values and values predicted by the line.

When examining a hypothetical sample of 61 penguin burrows, the question arises: is there indeed a linear relationship here? To support this assertion, we check whether the line fits the data well. The coefficient of determination, denoted as \(r^2\), provides evidence: with a value of 0.386, it indicates that approximately 38.6% of the variability in trail length can be explained by the variability in soil hardness. While \(r^2\) gives us a quick glance at the fit, to verify the linearity and to see if the trend is statistically significant, we'd perform a hypothesis test.

Hypothesis Test

A hypothesis test in the realm of simple linear regression is used to ascertain whether the observed relationship between the independent variable and the dependent variable is statistically significant or if it has arisen by chance. For penguin trail length and soil hardness, our test would revolve around determining the significance of the slope coefficient of the regression line, \( -1.4187 \).

To conduct this test, we calculate the t-statistic for the slope coefficient using the standard error of the slope and compare it against a t-distribution with \(n - 2\) degrees of freedom, where \(n\) is the number of observations, which in this case is 61. The resulting p-value helps us decide if we should reject the null hypothesis, usually framed as 'there is no relationship,' in favor of the alternative hypothesis, 'there is a relationship.'

If the p-value is smaller than the desired significance level (often set at 0.05), we have sufficient evidence to say the relationship is significant. In our scenario, a negative coefficient suggests that as soil hardness increases, the trail length decreases. If this relationship is statistically significant, it supports the claim that penguins build shorter trails in harder soil.

Prediction Interval

When we use simple linear regression not just to understand relationships but to make predictions, we want to know how reliable these predictions are. That's where prediction intervals come into play. They give us a range, within a certain level of confidence (usually 95%), in which we expect the true value of 'y' (in this case, trail length) to fall, given a certain 'x' value (soil hardness).

The formula for a prediction interval is \(\hat{y} \pm t*se*\sqrt{1+1/n+(x-\bar{x})^2/\sum(x-\bar{x})^2}\), where \(\hat{y}\) is the predicted trail length, \(t\) is the t-value for our confidence level, \(se\) is the standard error of the estimate, and \(\bar{x}\) and \(x\) are the mean and specified values of the independent variable, respectively.

For example, to predict the trail length when soil hardness is 6.0, we plug that value along with the other parameters into our prediction interval formula. This interval reflects both the accuracy of our model and the natural variability of the data, providing a more complete picture than a point estimate alone. Including prediction intervals in our prediction gives a better insight into the expected precision, especially when dealing with biological data where variability is naturally high.