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Kepler's Law of Planetary Motion The time it takes for a planet to complete its orbit around the sun is called the planet's sidereal year. Johann Kepler studied the relation between the sidereal year of a planet and its distance from the sun in 1618 . The following data show the distances that the planets are from the sun and their sidereal years. $$ \begin{array}{lcc} \text { Planet } & \begin{array}{l} \text { Distance from Sun, } x \\ \text { (millions of miles) } \end{array} & \text { Sidereal Year, } \boldsymbol{y} \\ \hline \text { Mercury } & 36 & 0.24 \\ \hline \text { Venus } & 67 & 0.62 \\ \hline \text { Earth } & 93 & 1.00 \\ \hline \text { Mars } & 142 & 1.88 \\ \hline \text { Jupiter } & 483 & 11.9 \\ \hline \text { Saturn } & 887 & 29.5 \\ \hline \text { Uranus } & 1785 & 84.0 \\ \hline \text { Neptune } & 2797 & 165.0 \\ \hline \text { Pluto* } & 3675 & 248.0 \end{array} $$ (a) Determine the least-squares regression equation, treating distance from the sun as the explanatory variable. (b) A normal probability plot of the residuals indicates that the residuals are approximately normally distributed. Test whether a linear relation exists between distance from the sun and sidereal year. (c) Draw a scatter diagram, treating distance from the sun as the explanatory variable. (d) Plot the residuals against the explanatory variable, distance from the sun. (e) Does a linear model seem appropriate based on the scatter diagram and residual plot? (Hint: See Section 4.3.) (f) What is the moral?

Step by step solution

01

- Compile Data

Collect and organize the given data for distances from the sun (x) and sidereal years (y) of each planet. These are: Mercury (36, 0.24), Venus (67, 0.62), Earth (93, 1.00), Mars (142, 1.88), Jupiter (483, 11.9), Saturn (887, 29.5), Uranus (1785, 84.0), Neptune (2797, 165.0), and Pluto (3675, 248.0).

02

- Calculate Regression Equation

Use the least-squares regression method to determine the regression equation. This involves calculating the slope \(b_1\) and the intercept \(b_0\) using the formulas: \[ b_1 = \frac{\sum (x_i - \overline{x})(y_i - \overline{y})}{\sum (x_i - \overline{x})^2} \] and \[ b_0 = \overline{y} - b_1\overline{x} \]

03

- Verify Normality of Residuals

Construct a normal probability plot of the residuals and visually inspect it to ensure that the residuals are approximately normally distributed.

04

- Test Linear Relationship

Perform a hypothesis test to confirm if a linear relationship exists between distance from the sun (x) and sidereal year (y). Use the null hypothesis \(H_0: b_1 = 0\) and alternative hypothesis \(H_1: b_1 e 0\). Calculate the t-statistic and compare it against the critical value at a chosen significance level.

05

- Draw Scatter Diagram

Plot the known data points on a scatter diagram by marking distances from the sun (x) on the horizontal axis and sidereal years (y) on the vertical axis.

06

- Plot Residuals

Plot the residuals on a graph with the explanatory variable (distance from the sun) on the x-axis and the residuals on the y-axis. Examine the spread and pattern of the residuals.

07

- Evaluate Linear Model

Analyze the scatter diagram and the residual plot. If the scatter plot shows a non-linear pattern or if the residuals show a trend or systematic structure, the linear model may not be appropriate.

08

- Moral of the Analysis

Sum up the observations. For instance, if the data follows a general pattern expected by Kepler’s law and the residuals are randomly distributed, it would imply the linear model is reasonable to describe the relationship.

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Key Concepts

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

least-squares regression

Least-squares regression is a statistical method used to determine the relationship between two variables by minimizing the sum of the squares of the differences between observed and predicted values. Typically, one variable is designated as the explanatory variable, and the other as the response variable. Here, the distance from the sun is our explanatory variable, and the sidereal year is the response variable.

The least-squares regression equation has the form: \( y = b_0 + b_1x \), where \( b_0 \) is the y-intercept, and \( b_1 \) is the slope. These coefficients are calculated using the given formulas: \[ b_1 = \frac{\begin{array}{c}\text{sum of products of deviation of } x \text{and } y \text{ from their respective means}\right)}{\begin{array}{c}\text{sum of squares of deviation of } x \text{from its mean}\right)} \ \ b_0 = \text{mean } y - b_1 \times \text{mean } x \].

By applying these formulas to our planetary data, we can derive the least-squares regression line, which helps us predict the sidereal year based on the distance from the sun.

scatter diagram

A scatter diagram, also known as a scatter plot, is a type of graph that shows the relationship between two variables by displaying data points plotted on a two-dimensional plane. The x-axis represents the explanatory variable, and the y-axis represents the response variable. 

For our problem, the x-axis would show the distance of each planet from the sun, while the y-axis would show their respective sidereal years. By plotting the given data points - such as Mercury (36, 0.24) and Venus (67, 0.62) - we can visually inspect the relationship between distance from the sun and sidereal year. If the points tend to cluster around a straight line, this suggests a linear relationship between the variables.

Scatter diagrams are crucial for identifying any trends, patterns, or outliers that might exist within the data. This visual examination helps in evaluating whether a linear model is appropriate for the given dataset.

linear relationship

A linear relationship implies that two variables change in a proportional manner such that their plot forms a straight line. Mathematically, this is expressed as \( y = b_0 + b_1x \), where changes in the explanatory variable \(x\) result in predictable changes in the response variable \(y\).

In the context of Kepler's Law of Planetary Motion, we investigate whether there is a linear relationship between the distance of a planet from the sun and its sidereal year. A hypothesis test can be conducted to confirm this relationship by examining the slope \( b_1 \).

The null hypothesis \( H_0 \) presumes \( b_1 = 0 \), suggesting no linear relationship, while the alternative hypothesis \( H_1 \) posits \( b_1 e 0 \), indicating a linear relationship. By comparing the calculated t-statistic against the critical value at a determined significance level, we can accept or reject the null hypothesis. A significant \( b_1 \) value reinforces that a linear relationship is present in the data.

Conducting such tests and plotting residuals provide further validation if the assumptions of linearity, normality, and equal variances hold, making the linear model suitable for our data.