A student runs an experiment six times in an attempt to...

A student runs an experiment six times in an attempt to obtain an equation relating two physical quantities $x$ and $y$. For $x=1,2,4,6,8,10$ units, the experiments result in corresponding $y$ values of $3,3,4,6,7,8$ units. Find and graph the following: (a) the least squares line; (b) the least squares quadratic polynomial; $(c)$ the interpolating polynomial. (d) Which do you think is the most likely theoretical model for this data?

Key Concepts

Least Squares Method

The least squares method is a statistical technique used to find the best-fitting line or curve to a set of data by minimizing the sum of the squares of the differences between the observed values and the values predicted by the model. This approach provides a way to quantify and reduce the error inherent in approximating complex data with a simpler model, and it is commonly used in linear and nonlinear regression analysis.

Polynomial Regression

Polynomial regression extends the concept of linear regression by allowing the model to be a polynomial of higher degree, such as a quadratic. This method fits a polynomial function to the data in a least squares sense, allowing for a more flexible model that can capture non-linear patterns in the data while still minimizing the overall error between the observed and predicted values.

Polynomial Interpolation

Polynomial interpolation involves constructing a polynomial of degree (n-1), where n is the number of data points, that passes exactly through each prescribed data point. This method guarantees an exact fit at the data values, though it can lead to issues such as overfitting or oscillatory behavior, especially with a high number of points or unevenly spaced data.

Model Selection in Data Fitting

Model selection is the process of choosing the most appropriate mathematical model to represent a set of data. It involves balancing the complexity of the model against its ability to accurately capture the underlying trends in the data while avoiding overfitting. The decision often considers factors such as the theoretical basis for the model, the behavior of the residuals, and the predictive performance of simpler models versus more complex ones.

Transcript

00:01 The data given in the problem can be written in the form of a table as x values are minus 6, minus 4, minus 2, 0, 246, 8, 10 and 12.

00:18 And the corresponding y values are 10 .7, 9, 7, 5 .4, 3 .5, 1 .7, minus 0 .1 .7, minus 0 .1, minus 0 .1, minus 1 .8, minus 3 .6 and minus 5 .3.

00:41 Now for this given data, if we first want to create a scatter plot, we will use the graphing device, which is here we will use the ti84 graphing calculator.

00:51 Here go to stat and select the edit option, which is the first option.

00:57 Here we will see two lists.

00:59 Actually we will see multiple lists, but we will use only l1 and l2 here.

01:03 In l1 we will input the x values and in l2 we will input the y values.

01:10 So as you can see we have entered the x and y values here in l1 and l2.

01:16 Now to view this, to view the scatter plot select zoom and the ninth option zoom stat.

01:26 Here we can see the scatter plot.

01:30 Now as we can see these data points follow almost a linear decay path or a linearly decreasing path so we will model this using a linear function so to create the linear model select stat select the right arrow and here we will find the fourth option linear regression so if you select four now before we calculate the model let us store it in as a given function so in the store option press vars go to the right arrow to find yvars select the first option function and select the first function y1 so now that we have selected y1 whatever model is created for this data will be stored as a function y1 and now go to calculate so now the calculator will give us the linear model now any linear equation can be written in in its general form as y equals a x plus b where a the slope of this given model is minus 0 .89 and b the y intercept is 5 .327 so this is the linear model for this given data.

02:49 Now to see the plot of the linear model again press zoom and then the ninth option zoom stat so here along with the scatter plot the calculator will now show us the graph of the linear model...

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