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Chapter 3: Problem 35

What's my line? You use the same bar of soap to shower each morning. The bar weighs 80 grams when it is new. Its weight goes down by 6 grams per day on the average. What is the equation of the regression line for predicting weight from days of use?

Short Answer

Expert verified
The equation is \( y = -6x + 80 \).

Step by step solution

01

Identify the variables

In this problem, we have two variables: the number of days, which we'll call \( x \), and the weight of the soap, which we'll call \( y \).
02

Determine the initial value

The initial weight of the soap, when \( x = 0 \) (day 0), is given as 80 grams. This is the intercept \( b \) of the regression line.
03

Determine the rate of change

The soap's weight decreases by 6 grams per day. This is the slope \( m \) of the regression line. Therefore, \( m = -6 \).
04

Write the equation of the line

The general equation for a line is \( y = mx + b \). Substitute \( m = -6 \) and \( b = 80 \) into the equation: \[ y = -6x + 80 \]This equation represents the regression line for predicting the weight of the soap based on the number of days of use.

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Key Concepts

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

Intercept
The intercept in a regression equation is where the line crosses the y-axis. In simpler terms, it's the starting value of the dependent variable when the independent variable is zero. Imagine it as the initial point you begin with. For the soap example, the intercept is represented by 80 grams. This means that on day zero, the weight of the soap is 80 grams. It is the baseline weight before the soap starts to be used. This point helps in understanding the regression equation, as it tells us where the line begins on the graph.
Slope
The slope of a regression line is a measure of how much the dependent variable changes when the independent variable increases by one unit. It shows the rate at which one variable influences another. Consider the soap example: the weight of the soap decreases by 6 grams every day. This decrement is expressed as the slope of the line, denoted by -6.
  • The negative sign indicates a decrease.
  • The number shows the amount of change per increase in the independent variable.
So, with every passing day, the weight of the soap decreases by 6 grams. Understanding slope is crucial because it gives us insight into the relationship between the variables.
Regression Equation
A regression equation is a mathematical expression used to predict the value of one variable based on the value of another. It's commonly written in the form \( y = mx + b \). Here's a breakdown:
  • \( y \) stands for the dependent variable, which we predict.
  • \( m \) is the slope, indicating the change rate.
  • \( x \) is the independent variable, representing the value that's being manipulated.
  • \( b \) is the intercept, or starting point.
In the case of the soap, our regression equation \[ y = -6x + 80 \]predicts the soap's weight after any number of days, showing how each day affects the remaining weight. This equation tells us everything we need to anticipate the soap's condition over time.
Variables in Statistics
Variables in statistics refer to the characteristics or properties that can take different values. They are the elements we observe and measure. In the context of regression, we usually focus on two types: independent and dependent variables.
  • Independent Variable: This is the variable that we manipulate or control. In the soap scenario, it's the number of days the soap is used, denoted by \( x \).
  • Dependent Variable: This is the outcome we observe or measure. For the soap, it is the weight left, represented by \( y \).
Understanding these variables is essential for constructing accurate regression models. By knowing what changes and what results from those changes, we can build a meaningful relationship that helps in forecasting future outcomes.

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Most popular questions from this chapter

In a scatterplot of the average price of a barrel of oil and the average retail price of a gallon of gas, you expect to see (a) very little association. (b) a weak negative association. (c) a strong negative association. (d) a weak positive association. (e) a strong positive association.

Hot dogs Are hot dogs that are high in calories also high in salt? The figure below is a scatterplot of the calories and salt content (measured as milligrams of sodium) in 17 brands of meat hot dogs. (a) The correlation for these data is r = 0.87. Explainwhat this value means. (b) What effect would removing the hot dog brand with the lowest calorie content have on the correlation? Justify your answer.

Least-squares idea The table below gives a small set of data. Which of the following two lines fits the data better: \(\hat{y}=1-x\) or \(\hat{y}=3-2 x ?\) Make a graph of the data and use it to help justify your answer. (Note: Neither of these two lines is the least-squares regression line for these data.) $$ \begin{array}{lrllrr} \hline x: & -1 & 1 & 1 & 3 & 5 \\ y: & 2 & 0 & 1 & -1 & -5 \\ \hline \end{array} $$

Big diamonds \((1.2,1.3)\) Here are the weights (in milligrams) of 58 diamonds from a nodule carried up to the earth's surface in surrounding rock. These data represent a single population of diamonds formed in a single event deep in the earth." $$ \begin{array}{cccccccccc}{13.8} & {3.7} & {33.8} & {11.8} & {27.0} & {18.9} & {19.3} & {20.8} & {25.4} & {23.1} & {7.8} \\ {10.9} & {9.0} & {9.0} & {14.4} & {6.5} & {7.3} & {5.6} & {18.5} & {1.1} & {11.2} & {7.0} \\ {7.6} & {9.0} & {9.5} & {7.7} & {7.6} & {3.2} & {6.5} & {5.4} & {7.2} & {7.2} & {3.5} \\\ {5.4} & {5.1} & {5.3} & {3.8} & {2.1} & {2.1} & {4.7} & {3.7} & {3.8} & {4.9} & {2.4} \\ {1.4} & {0.1} & {4.7} & {1.5} & {2.0} & {0.1} & {0.1} & {1.6} & {3.5} & {3.7} & {2.6} \\ {4.0} & {2.3} & {4.5}\end{array} $$ Make a graph that shows the distribution of weights of these diamonds. Describe the shape of the distribution and any outliers. Use numerical measures appropriate for the shape to describe the center and spread.

Will I bomb the final? We expect that students who do well on the midterm exam in a course will usually also do well on the final exam. Gary Smith of Pomona College looked at the exam scores of all 346 students who took his statistics class over a 10-year period." The least-squares line for predict- ing final-exam score from midterm-exam score was \(\hat{y}=46.6+0.41 x\) . Octavio scores 10 points above the class mean on the midterm. How many points above the class mean do you predict that he will score on the final? (This is an example of the phenomenon that gave "regression" its name: students who do well on the midterm will on the average do less well, but still above average, on the final.)
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