):Summary of Statistical Tests
Lab 2: Amino Acids and Protein Structure
If we sample and make repeated measurements of
some quantity, Y, we can calculate the mean value of Y (signified by
):
where, SY is the sum of all of the Y values measured in the experiment and n is the total number of measurements.
One commonly used measure of the variability of a quantity is called the variance (s2).
The
difference of each measurement of Y from the mean is squared (
),
these differences are then summed (S )
and divided by the total number of measurements minus one (n-1).
A second measure of the variability is called the standard deviation (s):
where s is the square root of the variance. Standard deviation is a more intuitive measure than variance for two reasons. One is that the units of standard deviation are the same as the units of the values themselves. Another reason is that in a normal distribution, 68% of samples fall in the range of one standard deviation above or below the mean. For these reasons, we will generally use standard deviation rather than variance in our work.
The standard error of sample means can be calculated from:
S.E. = s_
Ön
where s is the standard deviation of the samples values and n is the number of samples (or sample size).
Lab 3: Microbiology and Experimental Design
Because of the common acceptance of the 5% error rate (also known as the P<0.05 level of significance), means are commonly reported along with their 95% confidence intervals (or 95% CI). The 95% confidence interval is the range around the true mean within which one expects 95% of sample means to fall. The 95% confidence interval is related to standard error and for large sample sizes it can be estimated from standard error:
Upper limit = sample mean + (1.96)(SE)
Lower limit = sample mean – (1.96)(SE)
Calculation of the exact 95% confidence interval is more complex than these equations and beyond the scope of this course. However, it can be easily calculated by computer so we will simply use statistical software to determine it.
Lab 4: Enzyme Kinetics: Measurement of Succinate Dehydrogenase Activity
A more straightforward way to assess the difference between two means is a statistical test known as the t-test of means.
The heart of the t-test is the calculation of a statistic known as the "t value". (A "statistic" is a numerical quantity derived from samples that describes some characteristic of the samples or population.) The formula for the t value associated with two sample means is:
Where the Y's are the samples and
is a sample mean. s is the sample standard deviation
and n is the sample size. The subscript ones and twos indicate the first
group and the second group respectively. By convention, the larger mean is
assigned the value of 
in the equation above to avoid a negative value of t,
but some statistical software does not do this, and thus produces negative
values for t. In that case, simply take the absolute value of the listed
t.
Lab 1: Laboratory Techniques and Spectrophotometry
Using Excel to Perform a Linear Regression (Construct a Trendline)
We can generate a graph ("scatterplot") of our data using Microsoft Excel.
1. Open Excel using the icon on your desk top. Use two columns in the spreadsheet for your data. Column A should be concentration in mg/ml (x-axis) and column B is absorbance at 600 nm (y-axis).
2. Enter your data into the respective columns.
3. Use your cursor to select the data in columns A and B. Click on the Chart Wizard button (looks like a red, yellow and blue histogram).
4. Select XY (Scatter) as the chart type, and scatter (upper left option) as the chart subtype. Click on "Next".
5. Check the data range. All cells in your spreadsheet should be highlighted. Click on "Next".
6. Give your graph a title (optional). Label the x and y axes appropriately (units should be included where they exist - absorbance is unitless). Click on "Next".
7. You can select the graph alone or as an object "embedded" in your spreadsheet. Choose one of these options and click on "Finish". You should now see a graph of your data. Resize the graph to an appropriate size.
8. Add a "trendline" (i.e., the line of best fit). Move the cursor to a data point and click on the right mouse button. Select "Add Trendline" and click using the left mouse button.
9. Select Linear as the Regression type. Click on the Options tab. Select "Display equation on chart" and "Display R-squared value on chart". Click OK. You can move the equation around on the chart so that it is easy to read. Excel has performed a linear regression analysis of your data, i.e. it determined the best straight line that goes through the center of your data.
Lab 4: Enzyme Kinetics: Measurement of Succinate Dehydrogenase Activity
Performing a t-Test of Means Using JMP-IN
To perform a t-test of means, you need to set up a JMP-IN table with two columns. One column should have the data to be analyzed and the other should be set to "character" and should contain values that assign the row to a category. Note: the names assigning a row to a given category must be exactly the same. To avoid accidentally misspelling the category, it is best to type it in the first cell, then paste it into the other cells. The table below shows how the practice data on the web site is set up.
Select "Fit Y by X" from the Analyze menu to begin the analysis. To specify that a comparison of means should be done, the grouping variable needs to be specified as X and the data as Y. You can do this in the dialog box that pops up when you click the button.
After the X and Y variables have been specified and "Fit Y by X" has been selected, a graphical display of the data will appear. Drop down the analysis list (click on the red triangle at the left of the graph heading) and select "Means/Anova/t-Test" from the list. The screen will then look something like the figure below. The horizontal line at the center of the mean diamond shows the mean of each sample. The tops and bottoms of the diamonds represent the 95% confidence intervals (CI). As previously discussed, when the 95% CI of two groups do not overlap, the two groups are significantly different. However, if the 95% CI overlap slightly the groups may still be significantly different. JMP-IN puts lines on the mean diamonds to indicate how far the diamonds can overlap with the sample means still significantly different. However, these lines only apply if both groups have the same sample size (which is the case in this example, since both groups have 10).
The text at the bottom of the window gives the calculated values for the statistical test. The value listed under "t-Test" should be the same as the t value calculated by hand. DF is the degrees of freedom for the test: (10-1)+(10-1). The value listed under "Prob>|t|" is the probability (or P) value for the test. When performing the t-test by hand, it is not possible to know the P value, but simply to know whether it is more or less than 0.05 by comparing the calculated t value with the critical value in a table. However, the JMP-IN program has the algorithm to calculate the actual P value given a particular calculated t and number of degrees of freedom. To determine whether the groups are significantly different, one simply needs to see whether the listed value is less that 0.05. Use of a table of critical values is irrelevant here.
Linear Regression Analysis with JMP-IN
Run the JMP-IN program and open a new data table. Title the first column "concentration", then enter into the table the volume of mitochondrial suspension added (in ml) for reactions 1, 2, and 3. (Note that in this case the volume of suspension is a proxy for enzyme concentration, since the suspension was added to the same total volume in each trial.) Title the second column "velocity", and enter into the table the velocities you calculate in the problem set question 2.A. Drop down the Analyze menu and click on Fit Y by X. Select concentration as "X, Factor" and velocity as "Y, Response". Click OK. In the resulting graph window, click on the red triangle to the left of the "Bivariate Fit of velocity By concentration" heading to drop down the analysis menu. Click on "Fit Line".
There are three values from the analysis that are useful in assessing how well the data fit the predictions of Equation 4.
Lab 5: Infrared Gas Analysis of Photosynthesis and Respiration in Plants
Conducting a Paired t-test Using JMP-IN
1. Open a data table. This can be done several ways:
· Open a blank table (from file menu, from "New Data Table" on the JMPIN starter screen, or by clicking on the New Data Table button on the button bar) and then type in your own numbers.
· Obtain the data set from a drive or by downloading. Then open it from the file menu, from "Open Data Table" on the JMPIN starter screen, or by clicking on the Open button on the button bar.
· If you are accessing a file from the class web page, click on the link for the file. If you are in the BSCI 111 lab and everything works correctly, JMPIN should startup automatically and load the file. At a microcomputer lab on campus, you may need to save the file and open it from within the JMP-IN application.
2. Select "Matched Pairs" from the Analyze menu, or click on the shortcut button if it is visible
3. Select a column that you want to analyze from the list on the left of the dialog box. Then click on the "Y, Paired Response" button. Repeat with the other column, then click OK. The first column that you select will be subtracted from the second, so if you want a negative value when the treatment is less than the control, then select the control column first, then the treatment.
4. The results will
appear in a new window. The graph
shows whether the differences tend to fall above or below zero (0 on the Y
axis). The red horizontal line is the average difference and the dashed red
lines are the 95% confidence limits of the mean difference. To report the
results of the test, record the value of t (t-Ratio), degrees of freedom
("DF"), and the P value ("Prob …") for the one-tailed test that
is appropriate in your case. See the figure below to determine which P
value to use.
Lab 8: Recombinant DNA I
Construction of a Standard Curve
1. Retrieve your gel image from the course web site. Measure the distance traveled by each size standard band from its sample well. You can make your measurements from a printout of the gel, or measure them directly on the computer screen. However, you can not mix the two types of measurements because they are not at the same scale. In the same way, measure the distance of your unknown fragment band from its well and the distance of your cut pUC19 band from its well.
2. Make a Microsoft Excel table of size marker mass (in bp) and the distance traveled by that band. Construct a standard curve relating distance migrated (mobility in cm, on the x-axis) to size (in log10 of base pairs, on the y-axis) by plotting the points in an XY scatter plot (no lines connecting points). Click on the graph, then select "Add trendline" from the Chart menu. Under Options, check the box to display the equation on the chart. Use this best-fit equation to estimate the size of your unknown fragment and the pUC19. NOTE: because of uncertainty in measuring the distances, you can assume that your estimates of fragment sizes could be + or - 10% of the actual values. So your estimates of the fragment sizes will not agree exactly with the known value of pUC19 or the estimates of other students with the same unknown.