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Chapter 13: Problem 41

According to "Reproductive Biology of the Aquatic Salamander Amphiuma tridactylum in Louisiana" (Journal of Herpetology [1999]: \(100-105\) ), the size of a female salamander's snout is correlated with the number of eggs in her clutch. The following data are consistent with summary quantities reported in the article. MINITAB output is also included. \(\begin{array}{lrrrrr}\text { Snout-Vent Length } & 32 & 53 & 53 & 53 & 54 \\ \text { Clutch Size } & 45 & 215 & 160 & 170 & 190 \\ \text { Snout-Vent Length } & 57 & 57 & 58 & 58 & 59 \\\ \text { Clutch Size } & 200 & 270 & 175 & 245 & 215 \\ \text { Snout-Vent Length } & 63 & 63 & 64 & 67 & \\ \text { Clutch Size } & 170 & 240 & 245 & 280 & \end{array}\) The regression equation is \(\begin{array}{lrrrr}Y=-133+5.92 x & & & & \\\ \text { Predictor } & \text { Coef } & \text { StDev } & T & P \\ \text { Constant } & 133.02 & 64.30 & 2.07 & 0.061 \\ x & 5.919 & 1.127 & 5.25 & 0.000 \\ s=33.90 & \text { R-Sq }=69.7 \% & \quad R-S q(a d j)=67.2 \% & \end{array}\) Additional summary statistics are $$ \begin{aligned} &n=14 \quad \bar{x}=56.5 \quad \bar{y}=201.4 \\ &\sum x^{2}=45,958 \quad \sum y^{2}=613,550 \quad \sum x y=164,969 \end{aligned} $$ a. What is the equation of the regression line for predicting clutch size based on snout-vent length? b. Calculate the standard deviation of \(b\). c. Is there sufficient evidence to conclude that the slope of the population line is positive. d. Predict the clutch size for a salamander with a snoutvent length of 65 using a \(95 \%\) interval. e. Predict the clutch size for a salamander with snout-vent length of 105 .

Short Answer

Expert verified
a. The equation of the regression line for predicting clutch size based on snout-vent length: \(Y=-133+5.92x\)\n b. The standard deviation of \(b\): \(1.127\)\n c. Yes, there is sufficient evidence to conclude that the slope of the population line is positive.\n d. The clutch size for a salamander with a snoutvent length of 65 using a \(95 \%\) interval is \(251.2\)\n e. The clutch size for a salamander with snout-vent length of 105 is \(489.4\).

Step by step solution

01

Equation of the Regression Line

The equation of the regression line for predicting clutch size based on snout-vent length is already provided in the statement as \(Y=-133+5.92x\). So, the output for predicting clutch size based on snout-vent length would be \(Y=-133+5.92(65)\), and after calculation we get \(Y=251.2\).
02

Calculate the Standard Deviation

The standard deviation \(b\) of the slope coefficient is given in the MINITAB output as 1.127.
03

Analyzing the Slope of the Population Line

The p-value corresponding to the coefficient of the snout-vent length in the regression analysis is 0.000. P-value less than 0.05 indicates that there's a statistically significant relationship between the snout-vent length and the clutch size is present. This means that there's enough evidence to conclude that the slope of the population line is positive.
04

Predict the Clutch Size for a Salamander with a Snoutvent Length of 65

We use the regression equation to predict the clutch size for a salamander with a snout-vent length of 65. So, \(Y=-133+5.92(65)\). After calculating, the anticipated clutch size is \(251.2.\)
05

Predict the Clutch Size for a Salamander with a Snout-vent Length of 105

Using the same regression equation, the clutch size for a salamander with a snout-vent length of 105 is found by substituting \(X = 105\) into the equation: \(Y=-133+5.92(105)\). After calculating, the expected clutch size is \(489.4.\)

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Key Concepts

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

Predictive Modeling
Predictive modeling is at the heart of statistical regression analysis. In essence, it is a technique used to develop models or algorithms that can foresee upcoming outcomes based on historical data. This exercise focuses on predicting the clutch size of a salamander based on the snout-vent length. By using predictive modeling, we aim to establish a relationship between the salamander's physical attribute and its reproductive output.
The primary tool for predictive modeling in this example is the regression equation, which allows us to predict the clutch size, even for snout-vent lengths outside the range of the observed data. This provides valuable insights and helps in making data-driven decisions in biology and other fields. Predictive modeling is foundational in various spheres, from finance to medicine, indicating its versatile applicability.
Correlation
Correlation is a crucial concept in understanding the dependency between two variables, such as the snout-vent length and clutch size of the salamander in this scenario. It provides insight into whether changes in one variable might provoke changes in another. In this exercise, the correlation is suggested between these variables because as the snout-vent length increases, the clutch size seems to also increase.
To quantify this relationship, we often use a statistical measure known as the correlation coefficient, which ranges from -1 to 1. A positive value indicates a direct relationship, as in this example, whereas a negative value indicates an inverse relationship. Understanding the strength and direction of the correlation can help in making predictions and assessments regarding the variables in question.
P-Value
In statistical tests, the p-value plays a critical role in determining the significance of results. The p-value is the probability of observing the given result if the null hypothesis is true. In this exercise, the p-value corresponding to the snout-vent length's coefficient is 0.000. This statistic is crucial in assessing whether the observed correlation between snout-vent length and clutch size is due to chance or signifies a true relationship.
A common threshold for significance is 0.05. If the p-value is below this number, it suggests there is less than a 5% chance that the result is random, giving us strong confidence to reject the null hypothesis. In this exercise, with a p-value of 0.000, we can confidently assert that a significant positive slope exists between snout-vent length and clutch size.
Regression Equation
The regression equation is a fundamental tool in predictive analytics, used to predict the value of a dependent variable based on the independent variable. In our scenario, the regression equation provided is: \[ Y = -133 + 5.92x \] where \( Y \) indicates the predicted clutch size and \( x \) represents the snout-vent length. This equation establishes a linear relationship, allowing us to predict clutch size for any given snout-vent length.
To use the equation, simply replace \( x \) with the desired snout-vent length. For instance, to predict the clutch size for a snout-vent length of 65, you calculate \( Y = -133 + 5.92 \times 65 \). This equation aids in forecasting and decision making, demonstrating its value in statistical and practical applications.

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Most popular questions from this chapter

If the sample correlation coefficient is equal to 1, is it necessarily true that \(\rho=1 ?\) If \(\rho=1\), is it necessarily true that \(r=1 ?\)

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