Predicting Subway Fare With Pizza Cost A Regression Analysis

Hey guys! Today, we're diving into a fun and practical application of statistics: using regression analysis to predict subway fares based on pizza costs! Yep, you heard that right. We'll be crunching some numbers to see if there's a relationship between the price of a slice and the cost of a subway ride. This is a great way to see how economic indicators can be connected and how we can use data to make predictions. So, let's buckle up and get started!

The Scenario: Pizza vs. Subway Fare

Imagine you're a savvy city dweller trying to budget your expenses. You've noticed that the price of your favorite pizza slice seems to fluctuate, and you've also seen changes in subway fares over time. You start to wonder: is there a connection? Can the cost of a slice of pizza actually tell you something about the future cost of your commute? That's the question we're tackling today.

To explore this, we'll use a statistical technique called regression analysis. Regression analysis helps us understand the relationship between two or more variables. In our case, we want to see if there's a linear relationship between the cost of pizza (our predictor variable, often denoted as x) and the subway fare (our response variable, often denoted as y). Think of it like this: we're trying to draw a line through a scatterplot of data points that best represents the trend. This line, described by a regression equation, will help us predict subway fares based on pizza prices.

Now, you might be thinking, “Why pizza?” Well, the price of pizza can be seen as a proxy for the overall cost of living in a city. It reflects factors like ingredient costs, rent for restaurant space, and labor expenses. Subway fares, on the other hand, are influenced by factors like operating costs, infrastructure maintenance, and ridership levels. If these underlying economic factors affect both pizza prices and subway fares, then we might expect to see a correlation between the two.

Before we jump into the nitty-gritty calculations, let's talk about why this is useful. Understanding the relationship between pizza prices and subway fares can give us insights into economic trends. For example, if we find a strong positive correlation, it might suggest that both are sensitive to the same economic pressures, like inflation or changes in the cost of energy. Furthermore, being able to predict subway fares based on pizza prices could be helpful for budgeting and financial planning, especially for people who rely on public transportation for their daily commute. It's not just an academic exercise; it has real-world implications!

Setting Up the Regression Equation

Okay, let's get to the math! The core of our analysis is the regression equation. This equation is a mathematical representation of the line that best fits our data. It allows us to predict the value of one variable (the subway fare) based on the value of another variable (the pizza cost). The general form of a simple linear regression equation is:

y = a + bx

Where:

• y is the predicted subway fare (our dependent variable)
• x is the pizza cost (our independent or predictor variable)
• a is the y-intercept (the value of y when x is 0)
• b is the slope of the line (the change in y for every one-unit change in x)

Our goal is to find the values of a and b that best describe the relationship between pizza cost and subway fare. We'll use the data from our table to calculate these values. There are several methods we can use, but the most common is the least squares method. This method minimizes the sum of the squared differences between the actual subway fares and the subway fares predicted by our equation. In other words, it finds the line that gets as close as possible to all the data points.

To calculate a and b, we'll need some summary statistics from our data. Specifically, we'll need the means of both the pizza cost (x̄) and the subway fare (ȳ), as well as the standard deviations of both variables (sₓ and sᵧ) and the correlation coefficient (r) between them. The correlation coefficient, r, is a measure of the strength and direction of the linear relationship between two variables. It ranges from -1 to +1. A value of +1 indicates a perfect positive correlation (as pizza cost increases, subway fare increases), a value of -1 indicates a perfect negative correlation (as pizza cost increases, subway fare decreases), and a value of 0 indicates no linear correlation.

The formula for calculating the slope, b, is:

b = r * (sᵧ / sₓ)

And the formula for calculating the y-intercept, a, is:

a = ȳ - b * x̄

Once we have calculated a and b, we'll have our regression equation. We can then use this equation to predict the subway fare for any given pizza cost. For example, if we want to predict the subway fare when a slice of pizza costs $3, we'll simply plug $3 into our equation for x and solve for y.

Calculating the Regression Equation: A Step-by-Step Guide

Let's imagine we have the following data (this is just an example, you'll need to use your actual data from the table):

(Note: We'll focus on Pizza Cost and Subway Fare for our regression analysis. CPI is interesting but not directly used in this simple linear regression.)

Step 1: Calculate the means (averages) of x and y.

• Mean of x (pizza cost): x̄ = ($2.50 + $2.75 + $3.00 + $3.25 + $3.50) / 5 = $3.00
• Mean of y (subway fare): ȳ = ($2.75 + $3.00 + $3.25 + $3.50 + $3.75) / 5 = $3.25

Step 2: Calculate the standard deviations of x and y.

This is a bit more involved, but there are plenty of online calculators and spreadsheet functions that can help. Essentially, the standard deviation measures the spread of the data around the mean.

• Standard deviation of x (pizza cost): sₓ ≈ $0.387
• Standard deviation of y (subway fare): sᵧ ≈ $0.387

Step 3: Calculate the correlation coefficient (r).

The correlation coefficient measures the strength and direction of the linear relationship between x and y. The formula is:

r = Σ[(xᵢ - x̄)(yᵢ - ȳ)] / [(n - 1) * sₓ * sᵧ]

Where:

• Σ means