Demand A store manager wants to know the demand y for an...

Demand A store manager wants to know the demand y for an energy bar as a function of price $x$. The daily sales for three different prices of the product are listed in the table. $$\begin{array}{|c|c|c|c|}\hline \text { Price, x} & {\$ 1.00} & {\$ 1.25} & {\$ 1.50} \\ \hline \text { Demand, y} & {450} & {375} & {330} \\ \hline\end{array}$$ (a) Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the data. (b) Estimate the demand when the price is dollar 1.40 . (c) What price will create a demand of 500 energy bars?

Demand A store manager wants to know the demand y for an energy bar as a function of price $x$. The daily sales for three different prices of the product are listed in the table.
$$\begin{array}{|c|c|c|c|}\hline \text { Price, x} & {\$ 1.00} & {\$ 1.25} & {\$ 1.50} \\ \hline \text { Demand, y} & {450} & {375} & {330} \\ \hline\end{array}$$
(a) Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the data.
(b) Estimate the demand when the price is dollar 1.40 .
(c) What price will create a demand of 500 energy bars?

Key Concepts

Inverse Prediction

Inverse prediction involves using a regression model to determine the value of the independent variable that corresponds to a given value of the dependent variable. This technique is useful when the goal is to achieve a specific target outcome, allowing one to adjust the input variable so that the predicted output meets the desired criteria.

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. It fits a line to the data points in such a way that the sum of the squared differences between the observed values and the values predicted by the line is minimized. This method is commonly applied in various fields to understand trends and make predictions based on past data.

Least Squares Method

The least squares method is a standard approach in linear regression that minimizes the sum of the squares of the residuals (the differences between observed and predicted values). This method ensures that the regression line is the best possible fit for the data according to the criterion of minimizing error, thereby yielding reliable parameter estimates for the slope and intercept of the linear model.

Regression Analysis for Prediction

Regression analysis is not only used to describe the relationship between variables but also to predict future values of the dependent variable based on known values of the independent variable. Once a regression model is fitted to data, it can be used to estimate responses for inputs that were not originally observed, facilitating effective forecasting and decision-making.

Transcript

00:01 Now, in this problem, we're asked to, we're given some data for, you know, price and demand for, what are the energy bars.

00:10 So we're told that at a dollar apiece, we sell about $450 at a buck 25 apiece.

00:17 We sell $375 and a buck 50, we sell $330.

00:22 And they want us to use a linear regression to find a leaf squares.

00:31 Regression line for this so a linear fit and then estimate the demand when the price is a dollar 40 and what the demand what what price were created demand of 500 now what we see here is that we plot these things we just have three data points so there's really not more much more than a if we actually put fit a quadratic function through here it would actually by necessity go through every data point it would be a perfect fit but that doesn't necessarily mean you know it's accurate in between data points or if we extrapolate out here or out here.

01:05 And so interestingly, this one says, we're actually asked to extrapolate here because it's asked what price were we sell 500, which is actually out above this here.

01:16 So we're actually extrapolating this line.

01:19 And extrapolating data is really never, not really ever a good idea unless you're very certain that it's, you know, the trend is going to continue.

01:30 Now, so what i did, i'm going to look at a go watch problem 24 for the details of how i did this using numbers here on an ipad.

01:44 I was put in the data, and then i used the linear estimate function here and the index function to extract the data from that returns.

01:59 Out of here...

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