Question

Joan is concerned about the amount of energy she uses to heat her home.

The scatterplot shows the relationship between x = mean temperature in a particular month

and y = mean amount of natural gas used per day (in cubic feet) in that month, along with the regression line y^=1425−19.87x

a. Predict the mean amount of natural gas Joan will use per day in a month with a mean temperature of 30°F.

b. Predict the mean amount of natural gas Joan will use per day in a month with a mean temperature of 65°F.

c. How confident are you in each of these predictions? Explain your reasoning.

Short answer

Part (a) Joan will use an average of $828.90$cubic feet of natural gas per day in a month with a mean temperature of $30$degrees Fahrenheit.

Part (b) Joan will need $133.45$cubic feet of natural gas each day in a month with a mean temperature of $65$degrees Fahrenheit.

Part (c) As per the given parts (a) and (b) there is no surety that the prediction is correct.

Step-by-step solution

Part (a) Step 1: Given information

$x:$The average temperature for a given month.

$y:$In that month, the average amount of natural gas utilized each day.

The following is the OLS regression line:

$\stackrel{⏜}{y}=1425-19.87x$

Part (a) Step 2: Concept

The most common method for fitting a line to a scatterplot is least squares.

Part (a) Step 3: Calculation

With a mean temperature of $30°F$, the estimated average amount of natural gas Joan will need every day in a month can be computed as follows:

$\stackrel{⏜}{y}=1425-19.87\left(30\right)$

$=828.90$

Part (b) Step 1: Calculation

A mean temperature of$30°F$ the amount of natural gas Joan will need every day in a month may be computed as:

$\stackrel{⏜}{y}=1425-19.87\left(65\right)$

$=133.45$

Part (c) Step 1: Explanation

Using a regression line to make a forecast is just a mathematical calculation that may or may not be accurate. As a result, the value projected above may or may not be right or true. As a result, there is no guarantee that the prognosis is accurate.