91Ó°ÊÓ

Bi-lo Appliance Super-Store has outlets in several large metropolitan areas in New England. The general sales manager aired a commercial for a digital camera on selected local TV stations prior to a sale starting on Saturday and ending Sunday. She obtained the information for Saturday-Sunday digital camera sales at the various outlets and paired it with the number of times the advertisement was shown on the local TV stations. The purpose is to find whether there is any relationship between the number of times the advertisement was aired and digital camera sales. The pairings are: $$ \begin{array}{|lcc|} \hline \begin{array}{l} \text { Location of } \\ \text { TV Station } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Airings } \end{array} & \begin{array}{c} \text { Saturday-Sunday Sales } \\ \text { (\$ thousands) } \end{array} \\ \hline \text { Providence } & 4 & 15 \\ \text { Springfield } & 2 & 8 \\ \text { New Haven } & 5 & 21 \\ \text { Boston } & 6 & 24 \\ \text { Hartford } & 3 & 17 \\ \hline \end{array} $$ a. What is the dependent variable? b. Draw a scatter diagram. c. Determine the correlation coefficient. d. Interpret these statistical measures.

Step by step solution

01

Identify the dependent variable

The dependent variable is the one that changes in response to another variable. In this context, the digital camera sales, expressed in thousands of dollars, is the dependent variable since it is expected to vary with the number of times the advertisement was aired.

02

Constructing the Scatter Diagram

Create a two-dimensional graph to visually represent the relationship between the number of airings (independent variable on the x-axis) and sales (dependent variable on the y-axis). Plot each city as a point where the x-coordinate corresponds to the number of airings, and the y-coordinate corresponds to sales in thousands.

03

Calculating the Correlation Coefficient

The correlation coefficient (Pearson's r) measures the strength and direction of the linear relationship between two variables. Use the formula: \[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}} \]where \( n \) is the number of data points, \( x \) and \( y \) are the variables. Substitute the given data points and solve.

04

Interpreting the Statistical Measures

The value of the correlation coefficient (r) ranges from -1 to 1. If \( r \) is near +1, it indicates a strong positive relationship, meaning as the number of airings increases, sales tend to increase. If \( r \) is near 0, there is no linear relationship, and near -1, a strong negative relationship. Using the calculated \( r \), interpret the strength and direction of the relationship.

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Key Concepts

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

Dependent Variable

In the context of a study or experiment, the dependent variable is an essential concept. It refers to the variable that is being tested and measured. Often, this variable is expected to change as a result of variations in another variable, known as the independent variable. For the Bi-lo Appliance Super-Store exercise, the dependent variable is the "Saturday-Sunday digital camera sales," measured in thousands of dollars. This is because the store wants to determine if these sales figures change based on the number of times advertisements are aired on TV. By examining how sales fluctuate with different ad airings, businesses can understand the impact of their advertising efforts on consumer behavior.

Scatter Diagram

A scatter diagram, also known as a scatter plot, is a graphical representation that showcases the relationship between two quantitative variables. It helps visualize potential correlations between these variables by plotting data points on a Cartesian coordinate system. In creating a scatter diagram for the Bi-lo Appliance Super-Store's data:

• The x-axis represents the number of times the advertisement was aired (independent variable).
• The y-axis reflects the Saturday-Sunday sales figures in thousands of dollars (dependent variable).

Each city mentioned in the study is represented by a point on the graph, where its position corresponds to the pair of values for airings and sales. By observing the scatter diagram, one can quickly assess whether a pattern or trend exists, indicating a relationship between the number of ad airings and sales outcomes.

Correlation Coefficient

The correlation coefficient is a statistical measure that quantifies the strength and direction of a linear relationship between two variables. It is denoted by the letter "r" and can range from -1 to +1.To calculate the correlation coefficient in the exercise, we use the formula:\[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}} \]where:

• \( n \) is the number of paired data points.
• \( x \) represents the number of ad airings.
• \( y \) denotes digital camera sales in thousands.

By substituting the actual values from the collected data into this equation, one can compute the correlation coefficient. This computed value will provide insights into whether there is a meaningful relationship between the advertising frequency and sales.

Pearson's Correlation

Pearson's Correlation, often expressed as the correlation coefficient, is a standard method for assessing linear relationships between two continuous variables. It provides both magnitude and direction of the correlation.

• If the value of Pearson's correlation is +1, it suggests a perfect positive linear relationship, where increases in one variable consistently lead to increases in the other.
• A value of -1 indicates a perfect negative linear relationship, with one variable increasing as the other decreases.
• A value around 0 suggests no linear correlation between the variables.

This measure is crucial for Bi-lo Appliance Super-Store to determine the effectiveness of advertising on TV. After calculating the correlation coefficient, the strength and direction of the relationship between ad frequency and sales can be interpreted. This understanding helps the store assess the impact of its marketing strategies and decide on future advertising plans.