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Chapter 3: Problem 26

Here are some hypothetical data: $$\begin{array}{lllllll}\hline x & 1 & 2 & 3 & 4 & 10 & 10 \\\y: & 1 & 3 & 3 & 5 & 1 & 11 \\\\\hline\end{array}$$ (a) Make a scatterplot to show the relationship between \(x\) and \(y\) (b) Calculate the correlation for these data by hand or using technology. (c) What is responsible for reducing the correlation to the value in part (b) despite a strong straight-line relationship between \(x\) and \(y\) in most of the observations?

Short Answer

Expert verified
The presence of two points with the same \( x = 10 \) but different \( y \) values reduces the correlation.

Step by step solution

01

Create a Table of Values

First, identify the pairs of values from the given data. We have: - \( x = 1, y = 1 \) - \( x = 2, y = 3 \)- \( x = 3, y = 3 \)- \( x = 4, y = 5 \)- \( x = 10, y = 1 \)- \( x = 10, y = 11 \)This set of pairs will now be used to plot the scatterplot and calculate the correlation.
02

Plot the Scatterplot

On a graph, plot each pair of \( (x, y) \) values. This will give you a visual representation:- Point 1 at \( (1, 1) \)- Point 2 at \( (2, 3) \)- Point 3 at \( (3, 3) \)- Point 4 at \( (4, 5) \)- Point 5 at \( (10, 1) \)- Point 6 at \( (10, 11) \)Most points show a positive linear trend except for the points with \ x = 10 \ that show variation.
03

Calculate the Mean of x and y

Calculate the mean of both \( x \) and \( y \) values:\[ \text{Mean of } x = \frac{1 + 2 + 3 + 4 + 10 + 10}{6} = 5 \]\[ \text{Mean of } y = \frac{1 + 3 + 3 + 5 + 1 + 11}{6} = 4 \]
04

Calculate the Correlation Coefficient (Setting Up)

Use the formula for the correlation coefficient, \( r \):\[ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} \]Find the deviations from the mean for each data point and calculate their products.
05

Complete Correlation Calculation

Using the deviations calculated, find:- \(\sum{(x_i - \bar{x})(y_i - \bar{y})} = (-4)(-3) + (-3)(-1) + (-2)(-1) + (-1)(1) + (5)(-3) + (5)(7) = 36 \)- \(\sum{(x_i - \bar{x})^2} = 16 + 9 + 4 + 1 + 25 + 25 = 80 \)- \(\sum{(y_i - \bar{y})^2} = 9 + 1 + 1 + 1 + 9 + 49 = 70 \)Plug these into the correlation formula:\[ r = \frac{36}{\sqrt{80 \times 70}} \approx 0.48 \]
06

Analyze Correlation Results

The calculated correlation coefficient, approximately \( r = 0.48 \), is not particularly strong, indicating moderate linear relationship. However, the apparent anomaly is due to the \( x = 10 \) points; one correlates with \( y = 1 \) and the other with \( y = 11 \), which dilutes the overall linear pattern visible in the rest of the data.

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Key Concepts

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

Scatterplot
A scatterplot is a type of graph used in statistics to display the relationship between two variables. In our exercise, the scatterplot is a visual representation of the pairs of data points, namely \( (x, y)\). By plotting each pair on a Cartesian plane, students can better understand how the variables interact with each other. These points visually display their distribution and can help identify patterns or trends.
  • The points \((1, 1), (2, 3), (3, 3), (4, 5), (10, 1), (10, 11)\) are all plotted.
  • Most points, such as \((1, 1)\) to \((4, 5)\), reveal a positive linearity.
  • The points \( (10, 1)\) and \( (10, 11)\) stand out, suggesting variability at \(x = 10\).
A scatterplot is more than simply a collection of dots; it succinctly tells a story about our data by allowing an immediate grasp of trends and outliers.
Linear Relationship
A linear relationship suggests that there's a consistent, predictable connection between two variables. In mathematical terms, this means that changes in one variable correspond to changes in another, following a straight-line pattern when plotted on a graph.

In the context of our data, we observe that for most of the \(x\) values from 1 to 4, as \(x\) increases, \(y\) increases too, outlining a direct linear relationship. Such a pattern is indicative of a positive correlation, typical of a line with an upwards slope.
  • From \(x = 1\) to \(x = 4\), \(y\) values show an increasing trend.
  • This linear pattern is disrupted at \(x = 10\), where two widely different \(y\) values appear: 1 and 11.
Understanding these kinds of relationships is pivotal for data analysis, as it allows prediction and inference about one variable based on another.
Mean Calculation
Calculating the mean, or average, is a basic yet highly useful statistical procedure. It provides a central value for the dataset, giving a quick sense of the data's location. By summing all the data points and dividing by the number of points, the mean helps in understanding the overall trend.

For the given exercise, the means are calculated as follows:
  • For \(x\) values: \( \text{Mean of } x = \frac{1 + 2 + 3 + 4 + 10 + 10}{6} = 5 \).
  • For \(y\) values: \( \text{Mean of } y = \frac{1 + 3 + 3 + 5 + 1 + 11}{6} = 4 \).
Recognizing these average values helps understand the general tendency of the data in terms of its center or balance. This is crucial for detecting deviations and understanding whether the mean accurately reflects the dataset's behavior.
Data Analysis
Data analysis is the systematic study of data to uncover patterns or insights. In this exercise, we apply several analyses, from visualizing data through scatterplots to calculating core statistical measures like the correlation coefficient. Each step aids in portraying a deeper understanding of the dataset.

Some critical points in our analysis include:
  • Identifying deviations from mean values, which reveals data variability.
  • Calculating the correlation coefficient \(r\) sheds light on the strength and direction of the linear relationship between \(x\) and \(y\).
  • The anomalies at \(x = 10\) highlight the importance of recognizing outliers that can influence results.
The correlation coefficient, calculated as roughly 0.48, tells us that there is a moderate linear relationship overall. Considering the anomalies helps in explaining why this correlation isn't stronger. Effective data analysis requires careful attention to each part of the dataset and the potential influences on statistical results.

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Most popular questions from this chapter

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