Observation: Organisms have parts which can be measured in terms of length. Suppose you previously made some measurements on yourself. With those measurements in mind, you would now have a "human ruler." You could measure a centimeter at a time with one of your fingernails, and could measure longer lengths by "walking" with your thumb and index finger.
| Which of your fingernails comes closest to 1 cm in width? | Right Left | Thumb Index Middle Ring Pinkie |
| What is the length between your thumb tip and extended index finger tip? | cm |
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:
| Width across knuckles of: | left hand | cm | right hand | 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.
| 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! We might also wonder whether the results of your study can be generalized to the entire human population, for example.
Observation: You now know something about your own two hands. You also notice that not everyone in the room is the same size overall.
Question: In spite of different absolute body sizes, does everyone have hands of equal width?
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? | |
Experiment: Your instructor will compile your hand width data along with your classmates' and will give you a photocopy.
| 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 yours, the formulae are:
mean = o(x,ø) = f(isu(i=1,n,xi),n) standard deviation = r(f(bbc(isu(,,(xi2))) - (o(x,ø))2n,n-1))
| Mean Width (cm) | Standard Deviation (cm) | |
| Left hands in sample: | ||
| Right hands in sample: |
| T-table | |||
|---|---|---|---|
| Degrees of Freedom | Critical Levels | ||
| 0.10 | 0.05 | 0.01 | |
| 1 | 6.314 | 12.706 | 63.657 |
| 2 | 2.920 | 4.303 | 9.925 |
| 3 | 2.353 | 3.182 | 5.841 |
| 4 | 2.132 | 2.776 | 4.604 |
| 5 | 2.015 | 2.571 | 4.032 |
| 6 | 1.943 | 2.447 | 3.707 |
| 7 | 1.895 | 2.365 | 3.499 |
| 8 | 1.860 | 2.306 | 3.355 |
| 9 | 1.833 | 2.262 | 3.250 |
| 10 | 1.812 | 2.228 | 3.169 |
| 11 | 1.796 | 2.201 | 3.106 |
| 12 | 1.782 | 2.179 | 3.055 |
| 13 | 1.771 | 2.160 | 3.012 |
| 14 | 1.761 | 2.145 | 2.977 |
| 15 | 1.753 | 2.131 | 2.947 |
| 16 | 1.746 | 2.120 | 2.921 |
| 17 | 1.740 | 2.110 | 2.898 |
| 18 | 1.734 | 2.101 | 2.878 |
| 19 | 1.729 | 2.093 | 2.861 |
| 20 | 1.725 | 2.086 | 2.845 |
| 21 | 1.721 | 2.080 | 2.831 |
| 22 | 1.717 | 2.074 | 2.819 |
| 23 | 1.714 | 2.069 | 2.807 |
| 24 | 1.711 | 2.064 | 2.797 |
| 25 | 1.708 | 2.060 | 2.787 |
| 26 | 1.706 | 2.056 | 2.779 |
| 27 | 1.703 | 2.052 | 2.771 |
| 28 | 1.701 | 2.048 | 2.763 |
| 29 | 1.699 | 2.045 | 2.756 |
| 30 | 1.697 | 2.042 | 2.750 |
| 40 | 1.684 | 2.021 | 2.704 |
| 60 | 1.671 | 2.000 | 2.660 |
| 120 | 1.658 | 1.980 | 2.617 |
| ° | 1.645 | 1.960 | 2.576 |
Student's T-test
Calculate the t-statistic for our samples from the following formula:
t = f((o(x,ø)1-o(x,ø)2) r(n),r(s12+s22)) = ________
This formula assumes that n=n1=n2.
Ignore any negative sign carried by the t-statistic.
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:
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 |
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 will be significantly larger when they are soaked than beans which have been kept dry.
Experiment: A sample of beans was divided into two sub-samples. One sub-sample was placed in water, the other sub-sample was kept in dry conditions. Use a balance to its greatest precision to determine the weight of each of 10 beans from each sub-sample.
| Soaked Beans | Dry Beans |
| Is this really an experiment? | Yes No |
| If no, why not? | |
| If not an experiment, what is it? | |
| Mean weight of Wet Beans | g |
| Standard Deviation of the Mean | g |
| Mean weight of Dry Beans | g |
| Standard Deviation of the Mean | g |
Analysis: 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 the t-test, are the two means significantly different? | yes no | ||||
Decision:
| Based on the t-test, the hypothesis: "Soaking causes beans to expand" is: | rejected not rejected |
| Our hypothesis used the term "expand" and our prediction used the term "larger." In our experiment we tested the weight of the soaked beans. | |
| What weight adjective would describe the soaked beans? | |
Observation: Our soaked beans sure do seem larger than the dry beans, but how can we measure the volume of an oddly shaped living-bean?
Question: Does soaking beans cause them to expand?
Hypothesis: Soaking does not cause beans to expand. [note alternate!]
Prediction: If soaking does not cause beans to expand, then beans will not be significantly larger when they are soaked than beans left dry.
Experiment: Measure the volume of bean seeds by displacement of water in a graduated cylinder. Put exactly 14 ml of water in the graduated cylinder. Now slowly add beans until the water level comes just below the 25 ml mark; do not put in more than 10 beans. Calculate the volume per bean by dividing the total volume of beans added by the number of beans added.
| Soaked Beans | Dry Beans | |
| Final Liquid Level | ml | ml |
| Starting Liquid Level | 14 ml | 14 ml |
| Total Volume of Beans Added | ml | ml |
| Number of Beans Added | beans | beans |
| Volume per Bean | ml/bean | ml/bean |
The group of dry beans receiving no treatment is the ____________ group.
The group of soaked beans is called the ________________ group.
Analysis:
| Examining the volume per bean, there is a striking difference. | |
| Can we perform a T-test on these data? | Yes No |
| If No, why not? | |
| If we wanted to redo our volume measurements, how could we do them so that we could use a statistical test for our analysis? | |
| We will not make any further measurements, but perhaps we may satisfy our need for significance by recalling that scientists find 5% error acceptable. | |
| Calculate the ratio of the volume per soaked bean to the volume per dry bean. | |
| The soaked beans occupy | % of the volume of the dry beans |
| Is there at least a 5% difference between the beans? | Yes No |
Decision:
| Based on a displacement test, the hypothesis: "Soaking does not cause beans to expand" is: | rejected not rejected |
| Why did we choose to rewrite our hypothesis this time to its alternate "no effect" form? | |
| By having our hypotheses rejected, are we poor scientists? | Yes No |
| Why did we not have the option to "prove" any of our hypotheses? | |
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