# RANDOM SAMPLING

```TEACHER ANSWER KEY
RANDOM SAMPLING SURVEY – LESSON 1
Check one:
Pre-survey
Post-survey
Name:
Directions:
This survey is both a pre- and post- survey. Put a check mark at the top of this paper next to the survey you are doing
1. In a random sample, each member of the population has an equal chance of being selected.
a. True
b. False
2. Scientists collect random samples of marine microbes because it is impossible to collect each individual microbe.
a. True
b. False
3. Microbes make up _____________ of the ocean’s biomass.
a. less than 1%
d. greater than 90%
4. The major groups of life on Earth are:
a. plants and animals
b. Prokarya, fungi and viruses
c. Archaea, Eukarya, and Bacteria
d. insects, fish and birds
5. There are 10 beads in a bag, two of which are blue. If you only get to pick one bead (blindly) from the bag, what
is the chance that you will get a blue bead?
a. 1%
b. 2%
c. 10%
d. 20%
TEACHER ANSWER KEY to STUDENT WORKSHEET
Lesson 1: Which Microbe is the Most Abundant?
HYPOTHESIS: The most abundant type of microbe (bead) in the ocean (bag) is bacteria (answers will vary).
DATA COLLECTION: TABLE 1 (SAMPLE DATA) – Answers will vary
Table 1. Microbe Abundance, Based on a Sample of 40
1
A
Student
B
C
Coccolithophore Cyanobacteria
(Blue)
(Green)
1
0
2
0
3
1
0
6
2
Joshua
3
Kyle
4
Joe
5
Malia
6
7
8
9
10
11
Group Total
6
7
12
Group %
15 %
18 %
**Sum of percentages exceeds 100% due to rounding error
D
Archaea
(Pink)
2
3
0
0
E
Diatom
(Purple)
0
1
0
0
F
Bacteria
(Red)
3
2
4
2
G
Virus
(White)
4
2
2
2
H
Total
5
13 %
1
3%
11
28 %
10
25 %
40
102%**
10
10
10
10
DATA COLLECTION: TABLE 2 (POPULATION DATA) – All groups should have the same answers.
Table 2. Microbe Abundance in the Ocean
A
1
2
3
Population Total
Population %
B
C
Coccolithophore Cyanobacteria
(Blue)
(Green)
24
45
8%
15 %
D
Archaea
(Pink)
45
15 %
E
Diatom
(Purple)
6
2%
F
Bacteria
(Red)
75
25 %
G
Virus
(White)
105
35 %
H
Total
300
100 %
1. According to the data your group collected in Table 1, what are the two most abundant types of microbes?
Answers will vary by group. Check for consistency within each group. Reach a consensus with the class on
how to handle ties. For sample data shown in Table 1 above:
Species #1: Bacteria Percentage = 28 %
Species #2: Virus
Percentage = 25 %
2. According to the sample data that you personally collected in Table 1, what are the two most abundant types of
microbes? Answers will vary by student. Check that students calculated percentages correctly. (Hint: To
calculate the percentage of each type of “microbe” (bead) that was collected individually by a student, have
the student divide the number of microbes of each color that they personally collected by 10, then multiply by
100%. For Malia (in Table 1 above):
Species #1: Cyanobacteria Percentage = 60 %
Species #2: Bacteria &amp; Virus (tie) Percentage = 20 %
3. What are the two most abundant types of microbes in the ocean (Table 2)?
Answers should not vary by group, unless objects have been lost or miscounted.
Species #1: Virus
Percentage = 35 %
Species #2: Bacteria Percentage = 25 %
4. Using the STUDENT READING—Lesson 1: Introduction to Marine Microbes, list one interesting fact about each
of the two most common microbes found in the ocean (Table 2). Answers will vary.
Microbe #1: Up to 10 billion viruses can be found in a single liter of seawater.
Microbe #2: Bacteria are beneficial because they recycle nutrients.
5. How did the data from your individual sample (question 2) and your group’s sample (question 1) compare with
the data for the total population (question 3)? Is this what you would expect? Why?
Answers will vary. Generally (but not always!), the population will be more closely approximated by the
larger group sample than by an individual student’s sample.
6. How did you ensure that your sampling was random and not biased toward collecting any one type of microbe?
Main reasons are that students selected their samples without looking (so they didn’t choose their favorite
colors) and the objects were identical in shape, size, and weight (so each was equally likely to be selected).
Also, each student poured their samples back into the bag before collecting the next sample so the population
remained constant*. Students also mixed the objects before randomly selecting their ten objects.
*Note: The experimental design isn’t perfect! To improve the experiment, each student could replace each
item before selecting the next item (rather than returning all 10 items at once).
7. How could scientists estimate the relative abundance of microbes in the ocean off Ala Moana Beach Park?
They simply need to take a representative random sample. Since most microbes are invisible, a good way to
do this would be to take a random sample of sea water, analyze it for microbial species (e.g., by using a
powerful microscope), and then use that information to extrapolate the abundance of microbes in a particular
environment. There are also more modern, “high-tech” techniques to identify microbes such as DNA analysis
and flow cytometry. Remember, a liter of sea water contains billions of microbes. For more information on
microbes, visit the C-MORE website (http://cmore.soest.hawaii.edu).
TEACHER ANSWER KEY to STUDENT WORKSHEET
Lesson 3: Testing Hypotheses with Statistics
1. Fill in Table 1 below with the observed frequency (O) for the actual sample of 100 M&amp;M’s and the expected
frequency (E) that Mars, Inc. claims to manufacture for each candy color. To help you get started, the observed
frequency of the brown candies is 33, while the expected frequency is 30. Complete the data table below for all
colors, and give a descriptive title to the table.
Table 1. Expected vs. Observed Color Distribution for M&amp;M’s
Brown
Yellow
Red
Observed frequency (O)
33
26
21
Expected frequency (E)
30
20
20
Orange
8
10
Green
7
10
Blue
5
10
2. In order to visualize these data, make a bar chart or histogram to compare the observed vs. expected frequencies.
Use one color for the observed values and another color for the expected values. Label your axes and give the
graph a title. If you have computer access, you may do this in Microsoft Excel. To create this graph in Excel, have
students recreate Table 1 (above) in an Excel spreadsheet. Highlight all the columns in the table, and follow the
steps under MAKING AN EXCEL GRAPH in STUDENT INSTRUCTIONS – Lesson 2: Introduction to Excel. Start with
step #4 for Excel 2003 or step #3 for Excel 2007. Charts will vary (for an example, see below).
Expected
ColorDistribution
Distributionfor
for
Expectedvs.
vs.Observed
Observed Color
M&amp;M's
(based
M&amp;M’s
(basedon
onaasample
sample of
of 100)
100)
Percentage
35
30
25
20
Observed
15
Expected
10
5
0
Brown
Yellow
Red
Orange
Green
Blue
Color
ColorofofM&amp;M’s
M&amp;M's
3. How similar are the observed and expected frequencies? Do you think the Goodness-of-Fit test will reveal that the
observed sample data supports the company’s claim? Answers will vary. Once students gain experience
conducting statistical tests, they will realize that these values are actually quite similar (considering the
relatively small sample size of 100), which suggests the Goodness-of-Fit test will support the company’s claim.
4. Writing Statistical Hypotheses
a. Write a null hypothesis to test the claim that Mars, Inc. is making for the color distribution of M&amp;M’s.
Remember the null hypothesis is what you are attempting to prove or disprove and that it must be
expressed in terms of equality. The first part of the null hypothesis is written for you; complete the rest.
Let P= the population proportion of a given color
H0: Pbrown = 30% and Pyellow = 20% and Pred = 20% and Porange = 10% and Pgreen = 10% and Pblue= 10%
b. Write the alternate hypothesis. This is the hypothesis that must be true whenever the null hypothesis is
false.
H1: At least one of the above proportions is different from the claimed value.
5. Write out the formula for the Chi-Square Test Statistic and define all terms.
(O  E )
 
E
2
2
where = the chi-squared test statistic
∑ = the symbol for taking a sum
O = observed value
E = expected value
6. Calculate and fill in the values in Table 2. Brown is already done, as an example.
Table 2. Goodness-of-Fit Test Calculations for a Sample of 100 M&amp;M’s
Color
Observed
Expected
Category
Frequency (O)
Frequency (E)
O-E
Brown
33
30
3
Yellow
26
20
6
Red
21
20
1
Orange
8
10
-2
Green
7
10
-3
Blue
5
10
-5
(O-E)
9
36
1
4
9
25
2
(O-E)2
E
0.30
1.80
0.05
0.40
0.90
2.50
7. Use your answers to Questions 5 &amp; 6 above to calculate the 2 test statistic.
 2 test statistic = 5.95 (which is the sum of all values in the far-right column of Table 2)
8. Now that we have determined the 2 test statistic, we want to compare this value to the critical 2 value. To do
this, we will need to know the degrees of freedom.
a. Degrees of Freedom = 5
b. How did you determine the number of degrees of freedom? The number of degrees of freedom is one
less than the number of categories
9. Using the degrees of freedom determined above and the 0.05 significance level, look up the critical 2 value on a
Table of Chi-Square Probabilities (an abbreviated version of this table, up to df=10, is given below).
Table 3. Table of Chi-Square Probabilities
df
1
2
3
4
5
6
7
8
9
10
0.10
2.706
4.605
6.251
7.779
9.236
10.645
12.017
13.362
14.684
15.987
0.05
3.841
5.991
7.815
9.488
11.070
12.592
14.067
15.507
16.919
18.307
0.025
5.024
7.378
9.348
11.143
12.833
14.449
16.013
17.535
19.023
20.483
0.01
6.635
9.210
11.345
13.277
15.086
16.812
18.475
20.090
21.666
23.209
 significance levels
Critical Value of 2 = 11.070
10. Compare the 2 test statistic to the critical value, and circle the correct answers below:
The 2 test statistic is (larger, smaller) than the critical value, so we (reject, do not reject) the null hypothesis.
11. Interpret your results. The color distribution of our sample of M&amp;M’s, although not exactly the same as that
claimed by the manufacturer, is sufficiently close to support the manufacturer’s claim. The small differences are
within the range of what is expected. (Note: these results are based on the 0.05 significance level).
```