What is Science?

 

  1. As a brief introduction to this Integrative Science course, it may be useful to find out what you think science is. Well?? What is science??

 

 

  1. Is science fundamentally different from other disciplines that are part of human learning/education? If so, how?

 

 

  1. Is science necessarily useful to humankind? If not, should it be? What is basic scientific research? What is applied research?

 

 

  1. How does science impact you on a daily basis?

 

  1. Is the media (TV, newspapers etc.) generally a good source for scientifically sound information? Why or why not?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Rules for Significant Figures

 

To determine the number of significant figures in a number use the following 3 rules:

 

  1. Non-zero digits are always significant
  2. Any zeros between two significant digits are significant
  3. A final zero or trailing zeros in the decimal portion ONLY are significant

Example:  .500 or .632000 the zeros are significant

                 .006  or .000968 the zeros are NOT significant

 

 

For addition and subtraction use the following rules:

 

  1. Count the number of significant figures in the decimal portion ONLY of each number in the problem
  2. Add or subtract in the normal fashion
  3. Your final answer may have no more significant figures to the right of the decimal than the LEAST number of significant figures in any number in the problem.

 

 

For multiplication and division use the following rule:

 

  1. The LEAST number of significant figures in any number of the problem determines the number of significant figures in the answer. (You are now looking at the entire number, not just the decimal portion)

*This means you have to be able to recognize significant figures in order to use this rule*

      Example: 5.26 has 3 significant figures

                      6.1 has 2 significant figures

 

 

 

Please also see Appendix A pg 587 of your textbook

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Measurement Homework

 

Careful measurement underlies all of science.  In this lab, you are going to carry out some simple yet important measurements of the mass, volume, and density for some objects.

 

You may find the following formulas useful:

 

·        Volume of a block=

·        Volume of a cylinder=

The radius is half the diameter

·        Volume of a sphere =  

The radius is half the diameter

·        Density = mass/Volume

 

Note that in the formulas above the quantities you measure directly like length, radius, mass, etc. are denoted in italics while the quantities that are derived, like volume or density are written in regular type.  One possible exception is that Volume can also be measured directly by placing the object to be measured in a graduated cylinder containing a known volume of water and measuring the volume change.

 

1.      Measure the mass and dimensions of each of the following.  Be sure to tell the material of which the object is made.  Use the electronic scale to measure the mass.  Use the ruler to measure “straight” dimensions and use the calipers to measure the diameters of the cylinder and the sphere.  Show only significant figures.

 

Object

material

mass  

length

width

height

diameter

a. block

 

 

 

 

 

 

b. cylinder

 

 

 

 

 

 

 

c. sphere

 

 

 

 

 

 

 

2.      Calculate the volume and density for each of your objects.  Show your calculations and use of significant digits.

 

Object

mass

Volume

Density

a. block

 

 

 

b. cylinder

 

 

 

 

c. sphere

 

 

 

3.      Measure the mass of one object using both the balance and measure its volume by using displacement.  Calculate densities from each pair of masses and volumes.  Show the use significant figures.

Object

mass/scale

mass/balance

Volume/calculated

Volume/measured by displacement

 

 

 

 

 

 

 

 

 

 

 

 

            a. Which one you would choose as most accurate, both from the point of view of accuracy of    measurement as well as significant figures.  Discuss your reasons

 

 

 

 

 

 

 

4.      Discuss the following:

a.       Explain the difference between mass and weight

 

 

 

 

 

 

 

b.      Which does the scale measure?   Which does the balance measure?   Explain.

 

 

 

 

 

 

 

c.       Why does the displacement method work for measuring the volume of some objects?  Would it work for all objects?

 

 

 

 

 

 

Statistical Definitions and Formulas

 

Population is the entire group of individuals or objects that are considered.

 

Sample is a subset of the population whose characteristics are being analyzed with the intent of making a statement about the entire population.

 

Sample size, n=number of objects or individuals in the sample.

 

Statistic is a number associated with a sample for the purpose of describing some property of the sample.

 

 

 

Statistics

When a numerical measurement x is associated with each element of the sample, certain descriptive statistics can be calculated

 

Sample mean,  =is a measure of the central tendency of the numbers     

 

Sample variance,  is a measure of how widely dispersed are the numbers .

 

Sample standard deviation is a measure of the average difference between the sample mean  and each of the numbers .  The smaller this is the more closely clustered are the numbers.

 

Standard error of the mean,  is a measure of dispersion we could get if we took many different samples from one population, calculated the mean of each sample, and then used the sample means as our data points.

 

 

 

 

 

 

 

 

 

 

 

 

Formulas

 

Volumes:

 

·        Volume of a block=

·        Volume of a cylinder=

The radius is half the diameter

·        Volume of a sphere =  

The radius is half the diameter

·        Density = mass/Volume

 

In these latter formulas, use the appropriate number of significant digits of

 

 

Statistics:

 

Given a set of numbers   

 

sample size = n,

mean,  =                                                                                                                

 

variance,  

 

standard deviation  

standard error of the mean, .

 

Given two data sets A and B with means  and , sample sizes  and , and standard deviations  and , the

t-statistic is

 

 

 

 

 

                                               

Population Sampling Homework


Assume you capture and weigh a sample of Gila aardvarks in the Chiricahua Mountains of Southeastern Arizona. The weights (in kg) of the ten aardvarks follow:
21 19 17 18 18 22 23 21 20 16


1. Calculate the mean weight of the sample of aardvarks.
2. What is the range?
3. Calculate the variance (s2) of the sample. The variance is the sum of all individual deviations squared, divided by (n-1). In other words do the following:
            a. Determine the deviation of each data point (subtract the sample mean from                     each data point). Some of your values will be negative numbers, but that's OK.
            b. Square each individual deviation.
            c. Add all of the squared deviations together.
            d. Divide this sum of deviations squared by the sample size minus one (n-1). The                            product of this division is the sample variance.
4. Calculate the sample standard deviation (s) simply by taking the square root of the sample variance.
5. The standard error of the mean (SEM) can be calculated by dividing the standard deviation by the square root of the sample size. Calculate the SEM.
6. The reporting of descriptive varies among scientific disciplines, but a common way of reporting them (and the way we will report them in this class) is as follows:
Mean ± SEM, n = ?

Write your answer in the above format:_________________________________


7. What is the basic problem associated with a large sample size?

8. What is the basic problem associated with a small sample size?

9. How do you know if your sample size is large enough? Again, opinions vary among scientists. Many biologists use this rule-of-thumb: the sample is large enough if the SEM is less than 10% of the sample mean. Recall that sample size is in the denominator of the SEM, so the larger "n" is, the smaller SEM is. Based on this guideline, would you conclude that your sample of Gila aardvarks is large enough to provide you with a reasonable description of the entire population?

 

 

 

 

 

 

 

 

 

 

Data Entry and Descriptive Statistics Using SPSS

 

Label your columns:

  1. Click on the tab at the bottom of page labeled Variable View
  2. Under the column titled “Name” type the name of your column in the box and hit enter
  3. In the second column titled ‘Type’ leave as numeric

      - or click the gray box and choose string to accept categorical data (Ex. Male,             Female); then go to the last column “Measure” and choose ordinal from the      dropdown menu.

  1. Continue until all of the data columns you will need are properly formatted.
  2. To return to the data entry screen, click on the “Data View” tab at the bottom of the page

 

Data Entry for Descriptive Stats:

  1. Click on the first empty box in column 1, type in your first data point (or category) and hit enter
  2. Type in the next data point, hit enter and continue until all data for that column has been entered
  3. Repeat the above steps for all your other columns of data

 

Obtaining Descriptive Statistics:

  1. Click “Analyze” on the upper toolbar and you will see a list of analyses appear.
  2. Highlight “Reports” and another menu will appear.
  3. Click on “Case Summaries”.
  4. In the window that opens you will see a box on the left with a list of your column labels and 2 empty boxes on the right titled “Variable(s)” and “Grouping Variable(s)”.
  5. Click on your column label that contains measurement data in the left box, then click the arrow between the two boxes to move your column label to “Variable(s)”. Click on the column label containing the data groups, then click the arrow between the two boxes to move your column label to “Grouping Variable(s)”. Remember-categorical data go in the “grouping variable” box.
  6. Click on ‘Display Cases’ (below the left box) to uncheck this box; if you leave this checked, your printout will contain a copy of all your data.
  7. Click the bottom tab labeled ‘Statistics’.
  8. You will see 2 boxes, one labeled ‘Statistics’ and one labeled ‘Cell statistics’.  Click on the statistical tests you want performed in the left box and click the arrow to move that test over to the ‘Cell statistics’ box (e.g. mean, variance, etc.)  Continue until all tests you want performed are listed in the ‘Cell statistics’ box.  Click Continue.
  9. Click OK.
  10. A new window called Output will open; you can now edit the results, run more tests, create graphs, and print out results from this page. (see below for example)

 

 

Generating Graphs:

  1. Click on Graphs on the upper toolbar, then highlight Interactive and finally click on Scatterplot
  2. You will see your column labels in the left hand box, click and drag the column label to the correct axis (x-axis or y-axis)
  3. Click OK

 

 

 

 

Basic Information on the t-Test

 

Hypothesis: The hypothesis is a tentative explanation based on observations you have made.  Your observations may have been followed up with a search of the literature for more information before you develop your hypothesis.

 

Example: Men’s hands are larger than women’s hands OR adding fertilizer to a plant makes it grow better.

 

Null hypothesis:  The actual null hypothesis is a more formal statement of your original hypothesis.  The null hypothesis is usually written in the following form:  There is no significant difference between population A and population B. 

 

Example:  There is no significant difference in hand size between males and females.  OR  There is no significant difference in the growth of fertilized plants vs. unfertilized plants.

 

The reason we write it in this form is that scientists are basically skeptics and their goal is to prove a hypothesis false.  In fact, you can never really prove that a hypothesis is true.  In addition, the null hypothesis is used because it allows you to relate your calculations of the difference between the sample means to a standard of zero.  

 

The t-Test:  We use this statistical test to compare our sample populations and determine if there is a significant difference between their means. The result of the t-test is a ‘t’ value; this value is then used to determine the p-value (see below).

 

 If we cannot use a statistical test (doesn’t have to be a t-test) to determine whether a significant difference exists, then it becomes difficult to convince other scientists that your research is worth anything.

 

P-value: The p-value is the probability that ‘t’ falls into a certain range.  In other words this is the value you use to determine if the difference between the means in your sample populations is significant.  For our purposes, a p-value < 0.05 suggests a significant difference between the means of our sample population and we would reject our null hypothesis.  A p-value > 0.05 suggests no significant difference between the means of our sample populations and we would not reject our null hypothesis.

 

Types of t-tests:  There are two types of t-tests, the unpaired and paired t-test that we will use in this course.

 

            Unpaired t-test:  This type of t-test is used when you have independent samples.          In other words your samples are not directly related to one another.  Ex.: Index finger length between males and females.

 

            Paired t-test:  In this t-test your samples are related.  You collected data before and after some manipulation of your subjects.  Ex.: Pulse before and after 3 cups of coffee.

 

 

THE SCIENTIFIC METHOD

 

 

The scientific method is a systematic way of studying a problem. Use of this method is not limited solely to scientists. In fact, you have probably used the scientific method from time to time even though you were not aware of it. We can divide the scientific method into five broad categories of activity.

Identifying a Problem

 

A person using the scientific method must first identify a problem that needs solving. For example, a scientist may be interested in knowing whether aspirin can cure baldness in men. You as a non-scientist may be trying to decide which of two computers is better or should you take a “dietary supplement” to increase your health. These problems can also be approached in a non-scientific manner; non-science results when decisions are based on emotion, personal recommendations or television ads.

The research problem: “Can aspirin cure baldness?”

Formulating a Hypothesis

 

After a problem is identified within the scientific method, a hypothesis is formulated. The scientist interested in aspirin as a cure for baldness might formulate the following hypothesis: aspirin causes the regrowth of hair on completely bald heads.

Null hypothesis: “There is no significant difference in the number of hairs growing on the heads of treated or untreated men.”

Experimenting

 

Our baldness expert might design an experiment that would compare two groups of individuals so that they differed by only one factor. That difference would be whether or not they took aspirin.

Obtain 20 completely bald men. Divide them randomly into two equal-sized groups of 10 men each. One group of ten would serve as the treatment group. Send the treatment group home but tell them to take four aspirins each day for 3 months. The other group would serve as a control group—they would go home to live life as usual with no aspirin.  At the end of 3 months bring everyone back to the laboratory and count the hairs--if any--on each head.

Analyzing Data: the Results

 

The next step in the scientific process is data analysis. On the basis of such analyses you can either reject or accept the hypothesis that was formulated at the beginning of the experiment. The data listed below are those collected from the aspirin-baldness experiment.

 

 

 

 

 

Group 1.  Control: no aspirin

 

Number of hairs on head:

            3          6          14        2          5          7          19        30        1          2

 

Group 2.  Treatment: four aspirins each day

 

Number of hairs on head:

2          1          19        3          7          6          3          5          22        29

 

Does aspirin appear to affect hair regrowth? You can't really say until the data are analyzed.

The data analysis will produce results that produce statistics (characteristics related to these samples) that are interpreted as:

1. There is no significant difference in the two samples.

OR

2. The two samples are significantly different.

Conclusion

 

            The conclusion is a statement about what the data analysis says about the hypothesis.

 

1. If there is no significant difference in the two samples, the aspirin treatment has had no significant effect on hair growth. Accept the null hypothesis.

OR

2. If there is a significant difference in the two samples, the aspirin treatment has had a significant effect on hair growth. Reject the null hypothesis Aspirin does cause a regrowth of hair on completely bald heads.

 

 

 

 

 

 

 

 

Hypothesis Testing Using SPSS

Unpaired t-test:

  1. Under ‘Variable View’ label your columns as follows:
    1. Column 1= ‘Grouping Variable’ (Ex. Sex)
    2. Column 2= ‘Data Variable’ (Ex. Height)
  2. In ‘Data View’ enter your data as follows:                                                                               

Sex

Height

M

66

M

62

F

55

 

  1. Click ‘Analyze’ on the upper toolbar and highlight ‘Compare Means’
  2. Click ‘Independent Samples T Test’
  3. Click on your Grouping Variable (Ex. Sex) and click the arrow to put it into the Grouping Variable box
  4. Click the tab labeled ‘Define Groups’ and type in your 2 group names.  You need to type this exactly as it appears in your data column.  Click ‘Continue’
  5. Click on your Data Variable and click on the arrow to put this into the Test Variable box
  6. Click OK

 

Understanding the Output:

t-statistic
 

p-value
 

 

 

 

 

 

Paired t-test:

  1. Label your columns and enter your data as follows:

Before

After

1

10

6

12

4

8
  1. Click Analyze and highlight ‘Compare Means’ then click on ‘Paired Samples T Test’
  2. Click on the labels in the left box and move to ‘Paired Variables’ box on the right using the arrow.  YOU MUST HIGHLIGHT BOTH COLUMNS FIRST BEFORE CLICKING THE ARROW TO MOVE THEM TO THE ‘PAIRED VARIABLES’ BOX
  3. Click Ok

 

Understanding the Output: