91Ó°ÊÓ

How much should a healthy Shetland pony weigh? Let \(x\) be the age of the pony (in months), and let \(y\) be the average weight of the pony (in kilograms). The following information is based on data taken from The Merck Veterinary Manual (a reference used in most veterinary colleges). $$ \begin{array}{r|rrrrr} \hline x & 3 & 6 & 12 & 18 & 24 \\ \hline y & 60 & 95 & 140 & 170 & 185 \\ \hline \end{array} $$ (a) Make a scatter diagram and draw the line you think best fits the data. (b) Would you say the correlation is low, moderate, or strong? positive or negative? (c) Use a calculator to verify that \(\Sigma x=63, \quad \Sigma x^{2}=1089, \quad \Sigma y=650\) \(\Sigma y^{2}=95,350\), and \(\Sigma x y=9930 .\) Compute \(r .\) As \(x\) increases from 3 to 24 months, does the value of \(r\) imply that \(y\) should tend to increase or decrease? Explain.

Step by step solution

01

Create the Scatter Diagram

Plot each point \(x, y\) from the data on a graph where the x-axis represents the age in months and the y-axis represents the weight in kilograms. This visual representation helps to see the trend.

02

Draw the Line of Best Fit

Observe the scatter plot to estimate a line that best represents the slope of the data points. This line should have a balance of points above and below it, reflecting the overall direction of the data.

03

Estimate Correlation

Visually inspect the scatter plot. If the points closely follow a straight line, the correlation is strong. If they are more spread out, it is moderate or low. In this case, the points form a pattern suggesting a strong, positive correlation.

04

Calculate Correlation Coefficient

The correlation coefficient \(r\) is calculated using the formula \(r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n \Sigma x^2 - (\Sigma x)^2][n \Sigma y^2 - (\Sigma y)^2]}}\), with \(n = 5\). Plug in the given sums: \(\Sigma x = 63\), \(\Sigma x^2 = 1089\), \(\Sigma y = 650\), \(\Sigma y^2 = 95350\), and \(\Sigma xy = 9930\) to compute \(r\).

05

Interpret the Correlation Coefficient

Once calculated, \(r\) will show the strength and direction of the linear relationship. If \(r\) is close to 1, the correlation is strong and positive, indicating that as \(x\) increases, \(y\) tends to increase.

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Key Concepts

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

Scatter Plot

A scatter plot is a type of graph used to represent data points along two axes to show the relationship between two variables. In this context, we use it to show how a Shetland pony's age might relate to its weight.

• **Creating a scatter plot**: Begin by labeling the x-axis with the age of the pony in months (3, 6, 12, 18, 24) and the y-axis with the corresponding weights (60, 95, 140, 170, 185).
• **Plotting points**: Each pair of values \(x, y\) is plotted as a dot on the graph. For instance, at 3 months, the pony might weigh 60 kg, so you'd place a dot above the 3 on the x-axis and in line with the 60 on the y-axis.

The scatter plot allows us to visually analyze the data and notice any trends, such as detecting an upward, downward, or neutral pattern.

Best Fit Line

A line of best fit, or trend line, is a straight line drawn through the scatter plot that best represents the data. This line is used to illustrate the general direction or pattern of the data points.

• **Purpose**: It helps in understanding whether there is a relationship between the two variables, age and weight in this case, and predicts future values.
• **Drawing the line**: Ideally, the line should pass as close as possible to all of the data points. It should have approximately the same number of points above and below it.

This line makes it easier to see any deviations from the overall pattern and is valuable for making predictions about uncharted data.

Linear Relationship

A linear relationship between two variables means that they are proportionally connected, forming a straight line when plotted on a scatter plot. In simple terms, as one variable increases or decreases, the other does so in a predictable way.

• **Indicators of a linear relationship**: In the pony example, as age increases, weight should generally increase if the relationship is linear.
• **Evaluating strength**: The strength of the linear relationship is gauged through the correlation coefficient \(r\). An \(r\) value close to 1 or -1 indicates a strong linear relationship; in our case, a strong positive correlation would imply that the ponies often weigh more as they grow older.

Understanding linear relationships is crucial for predicting outcomes and ensuring that interpretations of the data are accurate.