91Ó°ÊÓ

In Exercise \(2.12,\) ConEX0319 \(\quad\) sumer Reports gave the prices for the top 10 LCD high definition TVs (HDTVs) in the 30 - to 40 -inch category. Does the price of an LCD TV depend on the size of the screen? The table below shows the 10 costs again, along with the screen size. $$\begin{array}{lcl}\text { Brand } & \text { Price } & \text { Size } \\\\\hline \text { JVC LT-40FH96 } & \$ 2900 & 40^{\prime \prime} \\\\\text { Sony Bravia KDL-V32XBR1 } & 1800 & 32^{\prime \prime} \\\\\text { Sony Bravia KDL-V40XBR1 } & 2600 & 40^{\prime \prime} \\\\\text { Toshiba 37HLX95 } & 3000 & 37^{\prime \prime} \\\\\text { Sharp Aquos LC-32DA5U } & 1300 & 32^{\prime \prime} \\\\\text { Sony Bravia KLV-S32A10 } & 1500 & 32^{\prime \prime} \\\\\text { Panasonic Viera TC-32LX50 } & 1350 & 32^{\prime \prime} \\\\\text { JVC LT-37X776 } & 2000 & 37^{\prime \prime} \\\\\text { LG 37LP1D } & 2200 & \text { 37" } \\\\\text { Samsung LN-R328W } & 1200 & \text { 32" }\end{array}$$ a. Which of the two variables (price and size) is the independent variable, and which is the dependent variable? b. Construct a scatterplot for the data. Does the relationship appear to be linear?

Step by step solution

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a. Identifying the Independent and Dependent Variables

The independent variable is the variable that can be manipulated or controlled, while the dependent variable is the variable that shows an outcome or changes in response to the independent variable. In this case, the screen size can be considered the independent variable, as TV prices may depend on the sizes. The price is the dependent variable since it depends on the screen size. Independent Variable: Screen Size Dependent Variable: Price

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b. Creating a Scatterplot and Analyzing Linearity

To create a scatterplot, plot the screen sizes on the x-axis and the prices on the y-axis. Observe the relationship between the two variables. If the points seem to form a straight line, we can say that there is a linear relationship between screen size and price. Price (y-axis) vs. Screen Size (x-axis): Points on the scatterplot: {(40, 2900), (32, 1800), (40, 2600), (37, 3000), (32, 1300), (32, 1500), (32, 1350), (37, 2000), (37, 2200), (32, 1200)} After plotting these points on a graph, it appears that there is a positive trend, but the relationship is not perfectly linear. However, to confirm our observation, we could perform a correlation analysis or a linear regression analysis to quantify the degree of linearity.

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Key Concepts

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

Scatterplot

A scatterplot is an essential tool in data visualization. It helps to showcase the relationship between two variables by using a graph with points plotted. Each point mirrors observations for two variables: one on the x-axis and the other on the y-axis. In the context of our exercise, we used screen size on the x-axis (independent variable) and price on the y-axis (dependent variable).

When creating a scatterplot, the placement of points illustrates how one variable might be affected by changes in another. Scatterplots are particularly effective because:

• They provide a clear visual of how data points are distributed.
• They help to easily identify trends, correlations, and outliers within the dataset.
• They allow a quick check on the nature of the relationship between variables, whether it is linear or non-linear.

While scatterplots are incredibly helpful, further statistical analysis such as correlation or regression might be necessary to understand the extent and nuances of relationships between variables.

Linear Relationship

A linear relationship between two variables means that one variable changes at a constant rate with respect to another. This is often represented by a straight line in a scatterplot. In our exercise concerning TV prices and screen sizes, it was important to ascertain if there was a linear relationship.

To determine linearity:

• Observe if the plotted points in the scatterplot align closely to a straight line.
• Look for a consistent trend where increases or decreases in the independent variable lead to corresponding changes in the dependent variable.

The term 'positive trend' here indicates that as the size of the TV screen increases, the price tends to increase. However, the data did not fall perfectly on a single straight line, hinting at some deviations from perfect linearity.

For more precision in evaluating this relationship, statistical measures like correlation coefficients can be used. A correlation analysis provides a value between -1 and 1, where numbers close to 1 or -1 indicate a strong positive or negative linear relationship, respectively.

Data Analysis

Data analysis involves inspecting, cleaning, and modeling data to extract useful insights and inform conclusions. In exercises like ours, the goal is to consider the relationship between variables such as price and size of TVs, and derive meaningful conclusions.

In our context, the analysis can be broken down into several steps:

• Data Collection: Gather the raw data, which in this case, is the prices and screen sizes of TVs.
• Data Visualization: Use tools like scatterplots to visualize and gain a preliminary understanding of relationships.
• Trend Analysis: Look for patterns or trends, such as a linear relationship between screen size and price.

Advanced analysis may include applying statistical techniques like regression analysis to further quantify relationships and make predictions.

Data analysis is a powerful process that turns data into actionable insights. It's essential in making informed decisions in varied fields, including economics, marketing, and electronics.