Now that we understand how to combine errors in order to report a 90%
confidence interval for a calculated value, we can proceed to investigate the procedure
for comparing a measured value with an accepted value, or two measured values with each
other, to determine whether they are equal. In other words, let's suppose that the area of
your block in Lab
1 was given as 134.0 +/- 0.3 cm^{2} and that your measurement was 133.4
+/- 0.4 cm^{2}. Can we say that there is a significant difference between these
two areas? To put the question another way, is the difference between these two figures
due to random effects only?

At first glance you might suspect that the two figures do not differ
significantly, since the difference between them is 0.6 cm^{2} and this difference
is less than the sum of the confidence intervals, which is 0.7 cm^{2}. Once
again, this simple conclusion is wrong for the same reason - the odds don't favor your
being too low on one measurement while simultaneously being too high on the other.
Theory tells us that to 90% confidence the difference between the two figures, if due to
random effects only, can be expected to be no larger than the square root of the sum of
the squares of the individual confidence intervals given above:

This is an example of using statistical methods to test a ** null
hypothesis. ** In the present case, our hypothesis might be stated in the
form of a question: Is

A_{accepted} - A_{measured}
= 0?;

that is, are the two areas different from one another. When we
subtract these two numbers, the presence of random error makes it unlikely that we would
get exactly zero. Another way to state this question is: Is the number 0 contained in the
appropriate 90% confidence interval when you subtract the two areas? For our example
(134.0 +/- 0.3 cm^{2}) - (133.4 +/- 0.4 cm^{2}) = (0.6 +/- 0.5 cm^{2})
does not include 0. In the present case, since the two values for the area differ by more
than can be accounted for due to random effects only, the conclusion that the two areas
are the same is __probably__ false.

Suppose that your two numbers for the area had differed from one another by an amount that could easily be attributed to random error, or even agreed with one another exactly. Would this "prove" the hypothesis that the areas are equal? No. The agreement might be only a statistical anomaly. Even if repeated experiments always agreed with one another within the interval expected due to random error, we would have to admit the possibility that continued improvement in the precision of the experiment might ultimately lead to detecting a statistically significant difference. This approach illustrates an important philosophical principle concerning experimental results. Although we can be reasonably sure (60% confident, 90% confident) that a given hypothesis is false (i.e, two numbers differ by more than we can account for due to random effects only), we can never prove with equal assurance that it is true.

In many experiments there is a linear relationship between the measured
variables. For example, the velocity of an object in free fall changes linearly with time,
in the absence of air resistance. When we plot a set of data and find that it approximates
a straight line, the next question is how to find the slope and the intercept of the line
which seems to provide the best fit. In last week's lab, our plot of average speed versus
time showed us an excellent example of how real data is scattered around a straight line.
We do not expect to find an __exact__ fit because we know that the presence of random
error causes this scatter away from an ideal straight line. We can find approximate values
for slope and intercept by using a straight edge to draw a line which seems to "split
the difference" between the scattered points. A more exact answer is given by the
Method of Least Squares, to which we now turn.

Just as in our earlier statistical procedures, the underlying model in this case is the normal, or Gaussian, distribution. We assume that, for each given value of x, repeated measurements of y would yield results distributed about some mean with some standard deviation. Although different mean values of y would be obtained for different choices of x, the model assumes that the standard deviation in the values of y is the same for every x. This assumption makes it possible for us to find the best fit even if we measure only one value of y for each value of x.

The process of finding this best fit proceeds as follows: If there were no random errors present, all of our experimental results y would fall exactly on a line given by the equation

For a given value of x (denoted by x_{i}), the value y that we
actually obtain (denoted y_{i}) will differ from the ideal (error-free) value of y
by an amount given by

Based on the mathematics of the Gaussian distribution that we discussed in the first lab, the probability that this value of y occurs is given by

.

A similar statement can be made for each of the y values we have
obtained. Now, in general, the overall probability for the occurrence of successive events
is given by the __product__ of the individual probabilities for each event. The
probability P that N measurements will yield the N experimental values of y that we have
obtained is

Our problem then is this: find values for the slope** ***m* and
intercept *b* which will make this probability as large as possible. In other words,
find m and b such that the values of y which we have obtained from our measurements
represent a set of values of y which is most likely to occur.

The maximum value of P will occur when the exponent has its minimum
value, i.e., __when the sum of the squares of the deviations of your measured points from
the fitted line is a minimum__. This minimum value may be found using standard
techniques from the differential calculus, and occurs when *m*** **and *b*
are given by

,

.

Pretty clearly, the computations involved in finding *m* and *b*
will get tiresome even for a small number of experimental points. And just as clearly, the
procedure can be carried out systematically by a computer. The use of the computer to find
the best-fit line has already been demonstrated for you. It is called a
"trend-line" in Excel.

In addition to computing the slope and intercept, Excel can perform the
mathematical operations to give estimates of the standard errors in the slope and the
intercept. These are given by the **linest** command described in the online help or
the Excel Hints sheet provided. Provided your computations are based upon at least five
data points, you may expect that the odds are approximately 9 out of 10 that the slope
computed from an infinite number of measurements will fall within 2 standard errors of the
slope you have obtained, and similarly for the intercept. (Since we have only __one__
estimate of the slope and __one__ estimate of the intercept, the standard error and the
standard deviation are identical.) Thus, to 90% confidence, your slope is **m " 2*Std Err of m** and your intercept is **b " 2* Std Err of b, **where m, b, and the standard error of
each are given by the **linest** command.

In order to use the computer program intelligently, keep in mind the following points:

- In your experimental procedure
alwaysfix one variable (x) then measure the other (y).

- You must enter a
t least three data pointsor the least squares procedure will not work. You needat least five data pointsto get approximate 90% confidence intervals in the slope and intercept.

- The x,y data you enter might not be simply the experimental values you have obtained. For example, when you were entering the data from last week's experiment to measure g, you entered for y the average velocity calculated from measured position and for x time found from using the data point number.

- No matter how wildly scattered your data may be, or even if the variables are not linearly related, the computer can always come up with a slope and an intercept which is a best fit in the least squares sense. It's a good idea to make at least a crude plot of your data to be sure that your chosen method of plotting does yield something reasonably close to a straight line. (The smaller the standard error in the slope, the closer your points come to fitting a straight line exactly.)