Problem 11 - Algebra 2 and Trigonometry

Question

In \(11-17 :\) a. Draw a scatter plot. b. Does the data set show strong positive linear correlation, moderate positive linear correlation, no linear correlation, moderate negative linear correlation, or strong negative linear correlation? c. If there is strong or moderate correlation, write the equation of the regression line that approximates the data. The following table shows the number of gallons of gasoline needed to fill the tank of a car and the number of miles driven since the previous time the tank was filled. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|}\hline \text { Gallons } & {8.5} & {7.6} & {9.4} & {8.3} & {10.5} & {8.7} & {9.6} & {4.3} & {6.1} & {7.8} \\ \hline \text { Miles } & {255} & {230} & {295} & {250} & {315} & {260} & {290} & {130} & {180} & {235} \\ \hline\end{array} $$

Step-by-Step Solution

Answer

The data shows strong positive linear correlation with a regression line: \( y = 30.96x + 7.76 \).

Step 1: Understand the Data and Set Up the Axes

In this step, we need to understand what each row in the table represents and how we'll set up our graph. The 'Gallons' serve as the x-values, and 'Miles' are the y-values. Prepare a scatter plot with gallons on the x-axis and miles on the y-axis.

Step 2: Plot the Data Points

Using the pairs of values from the table, plot each (Gallons, Miles) on the scatter plot. For example, the first pair is (8.5, 255), so plot this point on the graph.

Step 3: Assess the Correlation

Examine the scatter plot to determine the type of correlation. Look for patterns such as an upward trend for a positive correlation or a random spread for no correlation. In this case, the points appear to fit a positive trend.

Step 4: Determine the Strength of the Correlation

Since the data points are fairly close to each other and form a line with a positive slope, the correlation appears to be strong positive linear correlation.

Step 5: Compute the Regression Line

To find the equation of the regression line, we need to calculate the slope and intercept using the formula: - The slope (m) is calculated as: \[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \]- The intercept (b) is calculated as: \[ b = \frac{(\sum y) - m(\sum x)}{n} \]Using the given data:\( n = 10 \), \( \sum x = 80.8 \), \( \sum y = 2240 \), \( \sum xy = 18028 \), \( \sum x^2 = 674.62 \)Substitute these values to find the slope and intercept.Slope: \( m = \frac{10(18028) - (80.8)(2240)}{10(674.62) - (80.8)^2} \approx 30.96 \)Intercept: \( b = \frac{2240 - 30.96 \times 80.8}{10} \approx 7.76 \)Thus, the equation of the regression line is: \( y = 30.96x + 7.76 \)

Key Concepts

Understanding Linear Correlation

When we talk about linear correlation, we are referring to how closely two sets of data move in relation to one another. In this context, we are using gasoline gallons as our independent variable (x-axis) and miles driven as our dependent variable (y-axis).
When examining a scatter plot, a strongly linear correlation means that the data points form a line or being closely grouped along a straight line. - **Positive Linear Correlation**: As one variable increases, so does the other. The scatter plot will show an upward trend. - **Negative Linear Correlation**: One variable increases, and the other decreases, resulting in a downward trend.
In our case, observing the plot, there is a positive trend as more gallons of gas correlate with more miles driven. If the points form a clear line with a increasing slope, it indicates a strong positive linear correlation.

What is a Regression Line?

The regression line, or "line of best fit," is a straight line that best represents the data on a scatter plot. It serves as a predictor for the relationship between the two variables. The goal is to minimize the distance between the data points and the line itself.
A regression line helps in predicting the value of the dependent variable based on the independent variable. For instance, once we establish the regression line from our data about gasoline and miles, we can predict how many miles a certain amount of gallons might cover.

• It's calculated using the formula: \[ y = mx + b \]
• Where: \( y \) is the dependent variable (Miles driven), \( m \) is the slope, \( x \) is the independent variable (Gallons of gasoline), and \( b \) is the y-intercept.

This formula describes the best linear relationship between the variables and can be used to forecast future values.

Decoding Slope and Intercept

The slope and intercept are fundamental components of linear equations and are crucial in understanding regression lines.
- **Slope (m)**: This tells us the steepness of the regression line, and it indicates how much the dependent variable is expected to increase (or decrease) for a one-unit increase in the independent variable. In our example, a slope of approximately 30.96 means each additional gallon of gasoline results in about 31 more miles driven. - **Intercept (b)**: The intercept represents the value of the dependent variable when the independent variable is zero. In our scenario, an intercept of approximately 7.76 means that if no gasoline is used (hypothetically speaking), you'd start with the baseline miles, which might be explained by trip setup or short residuals.
In summary, both the slope and intercept help us understand the relationship's dynamics and provide a full equation used for making predictions and analysis of the data.

The Role of Data Analysis

Data analysis is the process of inspecting, cleaning, and modeling data to extract valuable information. It’s especially important in our scenario of using gasoline data to gain insights about driving habits or vehicle efficiency.

• **Scatter Plot Creation**: The initial step is to visualize the data points by plotting gallons against miles. This plot helps in observing any apparent trends or correlations directly.
• **Correlation Assessment**: After plotting, examining the scatter for patterns or trends reveals the type of correlation, providing an initial understanding of the data relationship.
• **Regression Calculation**: Finally, using data up to calculate a regression line, including slope and intercept, allows for quantifiable predictions and deeper analysis.

With these steps, data analysis can help us turn raw data into structured insights, paving the path for informed decisions or predictions. By breaking down each step, you can better understand the full picture and anticipate future trends in driving habits based on gasoline usage.