3. The method of linear least squares is frequently used to fit a straight line

Question

3. The method of linear least squares is frequently used to fit a straight line to a set of data, and is employed as part of a calibration curve analysis. Do all parts below. A. Concisely state all assumptions for using the method of linear least squares, using generally accepted symbols and/or equations if appropriate. B. What sample preparation precautions must be followed when preparing both the analyte and the standard samples? Include any special steps taken before or during measurement of all samples.

Explanation / Answer

In statistics, ordinary least squares (OLS) or linear least squares is a method for estimating the unknown parameters in a linear regression model, with the goal of minimizing the differences between the observed responses in some arbitrary dataset and the responses predicted by the linear approximation of the data (visually this is seen as the sum of the vertical distances between each data point in the set and the corresponding point on the regression line - the smaller the differences, the better the model fits the data).

ASSUMPTIONS:

Firstly, linear regression needs the relationship between the independent and dependent variables to be linear. It is also important to check for outliers since linear regression is sensitive to outlier effects.

Secondly, the linear regression analysis requires all variables to be multivariate normal. This assumption can best be checked with a histogram and a fitted normal curve or a Q-Q-Plot. Normality can be checked with a goodness of fit test, e.g., the Kolmogorov-Smirnof test.

Thirdly, linear regression assumes that there is little or no multicollinearity in the data. Multicollinearity occurs when the independent variables are not independent from each other. A second important independence assumption is that the error of the mean has to be independent from the independent variables.

Fourthly, linear regression analysis requires that there is little or no autocorrelation in the data. Autocorrelation occurs when the residuals are not independent from each other. In other words when the value of y(x+1) is not independent from the value of y(x). This for instance typically occurs in stock prices, where the price is not independent from the previous price.

The last assumption the linear regression analysis makes is homoscedasticity. The scatter plot is good way to check whether homoscedasticity (that is the error terms along the regression are equal) is given.

B.

sample preparation precautions:

1. Make sure the analyte and standard solution are contamination free.

2. As the amount to be measure for preparation of standard solutions and analyte is very small, it shuold be done carefully.

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