Statistics

A 1996 Sample Abstract is available.

In this exercise you will gain some experience with data collection, analysis, and basic decision making in science. Math is the foundation of these essential and basic procedures of science as a process. Since plant physiology is one subdiscipline of science, these fundamental methods are critical to future exercises where numerical data are generated. It is hoped that you will begin to understand why the word "prove" is not an important part of the scientific vocabulary. Outcomes in the biological world are influenced by environmental factors, but also by chance events; for this reason some facility with probability and statistics is essential to becoming a scientist.

Question: Is my right hand the same size as my left hand?

Hypothesis: My hands are the same size.

Prediction: If my hands are the same size, then the width across my left hand should be the same as the width across my right hand.

Experiment:Measure both hands across the knuckles:

Right ________cm Left ________cm

Was this really an experiment?    Yes      No

If no, why not? ___________________________________________

___________________________________________

If this is not an experiment, what is it? _____________

Analysis:

The width of my left hand is:

wider than the same as narrower than

that of my right hand.

Decision:

The hypothesis: "My hands are the same size" is:   rejected     not rejected

There is very little doubt about the outcome here because you have asked a discrete question with a measurable answer. Think about sources of error.

Did the prediction thoroughly test the hypothesis?     Yes       No
If not, what else might we measure to more thoroughly test the hypothesis?
(hint: the key word is "size"!)

1. _______________________________________

2. _______________________________________

Most investigations yield not only answers but more questions as well. Scientists are curious people! Can this simple observation of one individual be meaningful? Can the results of this study can be generalized to the entire population? Do you people-watch? If so, did you ever measure?

Question: Does everyone have equally wide hands?

Hypothesis: The human population has hands of equal width.

Prediction: If the human population has hands of equal width, then a sample of the human population should have hands of equal width.

Notice that we cannot go out and measure the hands of the entire human population, so we must settle for a sample. We hope we can take a representative sample (that is a random sample). Our sample will be all the people in this laboratory.

Would this be a random sample of the population?     Yes       No

If it is not a random sample, why isn't it?_________________________________

_________________________________________________

We also hope that our sample is sufficiently large. In spite of any shortcomings in our sample, we will continue our analysis since we lack a better sample.

Experiment: Hand width data for the class is posted on the board.

By collecting lots of data, do we now have an experiment?     Yes       No

If no, why not?________________________________________________

________________________________________________

If this is not an experiment, what is it? ______________

Analysis: Clearly we have various widths in each sample and must now include an assessment of this variation in preparing for our decision. Calculate the mean (average) width and the standard deviation of the samples. The latter gives us some measure of the variation (or spread) around the mean. Most calculators will determine the mean and standard deviation for you, but if not the formulae are:

mean =     standard deviation =

Right hands in sample:

Mean Width ______ cm ± Standard Deviation ______ cm

Left hands in sample:

Mean Width ______ cm ± Standard Deviation ______ cm

Scientists have developed methods for statistical testing of the data which give estimates of how much error is involved in the analysis. These fall into two categories: parametric and non-parametric tests. Parametric tests make one important assumption (among others): that our samples are from a normal distribution. This assumption can be justified graphically for each sample by plotting a histogram of the number of hands at a particular size (on the y-axis or ordinate) versus some equal size ranges (on the x-axis or abscissa). The plot should give a bell-shaped curve if the distribution is normal. Moreover, 68% of the data should fall within ± one standard deviation of the mean, 95% of the data should fall within ± two standard deviations of the mean, and 98% of the data should fall within ± three standard deviations of the mean.

Make the two essential histograms below and label the axes carefully!

 

Mean - SD = ______ Mean + SD = ______	Mean - SD = ______ Mean + SD = ______
Within 5% of 68%?     Yes       No	Within 5% of 68%?     Yes       No
Mean - 2xSD = ______ Mean + 2xSD = ______	Mean - 2xSD = ______ Mean + 2xSD = ______
Within 5% of 95%?     Yes       No	Within 5% of 95%?     Yes       No
Does it appear that our samples came from a normal distribution?     Yes       No

If our samples did not come from a normal distribution we would have to use a non-parametric test as we shall see below.

Calculate the t-statistic for our samples from the following formula:

Note: in cases where n₁=n₂:

Ignore any negative sign carried by the tstatistic.
This t-statistic is compared with a value in a t-table.

The table value is found by using a row defined by the degrees of freedom {in our case=n₁+n₂-2} and a column defined by the acceptable level of error. As scientists, what level of error is acceptable? Most scientists admit that 5% of the time chance alone will explain errors. Our table includes a 5% column.

Circle the pertinent table value in the table---------------------->

Decision Rules:

If the t-statistic is greater than the table value, the two samples are significantly different.

If the t-statistic is less than or equal to the table value, then the samples are statistically the same.

Decision: Based upon Student's T-test, the hypothesis:

"The human population has hands of equal width" is:   rejected       not rejected

Without much thought, we might have rejected our hypothesis if we found even one individual human with unequal hand size, but what hypothesis would we be testing if we did that?

Observations: A single bag of beans was purchased from the store. Some of the beans were soaked in water overnight, the rest from the same bag remain dry. Clearly the soaking has had some effect upon length.

Question: Does soaking beans cause them to expand?

Hypothesis: Soaking causes beans to expand.

Prediction: If soaking causes beans to expand, then beans which are soaked will be significantly larger than beans which have been kept dry.

Experiment: A sample of 20 beans was divided into two sub-samples. One sub-sample of 10 was placed in water, the other sub-sample of 10 beans was kept in dry conditions. Use a balance to its greatest precision to determine the weight of each bean.

Soaked
Dry

Is this really an experiment?   Yes       No

Analysis:

Mean weight of Wet Beans ______ g ± Standard Deviation of the Mean ______ g

Mean weight of Dry Beans ______ g ± Standard Deviation of the Mean ______ g

Carry out a t-test to see whether there is any significant difference between the two sub-samples.

T-statistic: ________ Degrees of Freedom: _________ Table Value: ________

Based on a t-test the soaked beans are:   heavier   lighter   the same weight

Go on to the Next Part!