91Ó°ÊÓ

The table below shows the number of cars (in millions) sold in the United States for various years and the percent of those cars manufactured by GM. $$ \begin{array}{|lcc|ccc|} \hline \text { Year } & \text { Cars Sold (millions) } & \text { Percent GM } & \text { Year } & \text { Cars Sold (millions) } & \text { Percent GM } \\ \hline 1950 & 6.0 & 50.2 & 1985 & 15.4 & 40.1 \\ 1955 & 7.8 & 50.4 & 1990 & 13.5 & 36.0 \\ 1960 & 7.3 & 44.0 & 1995 & 15.5 & 31.7 \\ 1965 & 10.3 & 49.9 & 2000 & 17.4 & 28.6 \\ 1970 & 10.1 & 39.5 & 2005 & 16.9 & 26.9 \\ 1975 & 10.8 & 43.1 & 2010 & 11.6 & 19.1 \\ 1980 & 11.5 & 44.0 & 2015 & 17.5 & 17.6 \\ \hline \end{array} $$ Use a statistical software package to answer the following questions. a. Is the number of cars sold directly or indirectly related to GM's percentage of the market? Draw a scatter diagram to show your conclusion. b. Determine the correlation coefficient between the two variables. Interpret the value. c. Is it reasonable to conclude that there is a negative association between the two variables? Use the .01 significance level. d. How much of the variation in GM's market share is accounted for by the variation in cars sold?

Step by step solution

01

Creating the Scatter Diagram

Open your statistical software and input the data for the years, cars sold, and GM percentage. Use a scatter plot tool to represent 'Cars Sold (millions)' on the x-axis and 'Percent GM' on the y-axis. Observe the trend shown by the scatter plot to infer the relationship between the number of cars sold and the percentage sold by GM.

02

Calculate the Correlation Coefficient

Using the software, calculate the correlation coefficient (often denoted by \( r \)) between the two variables: 'Cars Sold' and 'Percent GM'. The correlation coefficient will tell us the strength and direction of the linear relationship between these variables.

03

Analyzing the Correlation Coefficient

If \( r \) is close to -1, it suggests a strong negative relationship. If \( r \) is close to 1, it suggests a strong positive relationship. If \( r \) is near 0, it suggests no linear relationship.

04

Test for Significance

Perform a hypothesis test for the correlation coefficient at the 0.01 significance level. The null hypothesis (\( H_0 \)) is that there is no correlation, while the alternative hypothesis (\( H_a \)) is that there is a negative correlation. Calculate the test statistic and compare it to the critical value from the correlation table for a significance level of 0.01. If the test statistic is beyond the critical value, reject \( H_0 \).

05

Coefficient of Determination

Calculate the coefficient of determination (\( R^2 \)) by squaring the correlation coefficient \( r \). This value represents the proportion of variation in GM's market share explained by the variation in cars sold.

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Key Concepts

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

Scatter Diagram

A scatter diagram is a graphical tool used to visualize the relationship between two numerical variables. For this particular exercise, the scatter diagram helps us understand the relationship between the number of cars sold and the percentage of cars sold by GM. By plotting these variables on a Cartesian plane, with 'Cars Sold (millions)' on the x-axis and 'Percent GM' on the y-axis, you can observe any visible patterns or trends. Common patterns you might see include:

• Positive Association: As one variable increases, the other also increases.
• Negative Association: As one variable increases, the other decreases.
• No Apparent Association: The points do not seem to form any pattern.

In this scenario, observe how the plotted points are distributed. A downward sloping trend may indicate a negative correlation, where an increase in cars sold corresponds to a decrease in GM’s market share. This visual representation is the foundational step in correlation analysis.

Correlation Coefficient

The correlation coefficient, often denoted as \( r \), quantifies the strength and direction of a linear relationship between two variables. Its values range from -1 to 1.Here’s how to interpret the coefficient:

• \( r = 1 \): Perfect positive linear relationship.
• \( r = -1 \): Perfect negative linear relationship.
• \( r = 0 \): No linear relationship.

For the exercise at hand, the correlation coefficient tells us how strongly car sales are related to GM's market share. A negative \( r \) value would suggest that as the number of cars sold increases, the percentage of cars sold by GM decreases. To derive \( r \), input your data into statistical software. It uses the formula:\[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2-(\sum x)^2][n\sum y^2-(\sum y)^2]}}\]Understanding \( r \) gives insight into the linear dynamics between variables, which is crucial for making predictions or decisions based on this relationship.

Coefficient of Determination

The coefficient of determination, denoted as \( R^2 \), indicates the proportion of variance in one variable that can be predicted from the other variable. In simpler terms, it tells us how well the data fits the regression line. Calculated by squaring the correlation coefficient \( r \), \( R^2 \) ranges from 0 to 1:

• \( R^2 = 1 \): Perfect fit, the regression line explains all the variability in the data.
• \( R^2 = 0 \): The regression line does not explain any of the variability.

In our exercise, \( R^2 \) explains how much of the variation in GM's market share can be accounted for by the variation in cars sold. For instance, an \( R^2 \) of 0.64 would mean 64% of the variation in GM's market share is explained by car sales, leaving 36% due to other factors. Grasping \( R^2 \) is vital for evaluating the effectiveness of predictive models used in correlation analysis.

Hypothesis Testing

Hypothesis testing in correlation analysis helps us determine if the observed correlation is statistically significant. For relationships like the one between car sales and GM's market share, hypothesis testing affirms whether a negative correlation holds true in the broader population.The process involves:

• Null Hypothesis \( (H_0) \): Assumes no correlation between the variables.
• Alternative Hypothesis \( (H_a) \): Assumes a negative correlation exists.

Using the 0.01 significance level, calculate the test statistic from the correlation coefficient and compare it to the critical value:\[ t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}\]If the test statistic exceeds the critical value, you reject \( H_0 \), supporting the case for a significant negative association.Hypothesis testing is crucial for validating the correlation results, ensuring that the observed patterns are not due to random chance. This step aids in making informed decisions based on sound statistical reasoning.