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IB Biology / IHS
Just My Cup of “t”! (Test, that is!)
Or: How to Conduct an Unpaired Two-Tailed t-Test
Adapted from Jennifer Lockwood, Newbury Park High School, Newbury Park, CA
Introduction
What is statistics? Statistics is the science of collecting,
organizing, analyzing, and interpreting data in order to
make decisions. There are many different types of statistical tests—
for example, the Chi-Squared Test, the Pearson Correlation Test, or ttests. A t-test examines the difference between the means of two
sets of data.* This is done in order to determine if any observed
difference is due to chance alone or if the difference between the means
is the result of some other factor. If the difference is the result of some
other factor, the difference is said to be significant. (Remember,
significant means that the results obtained did not happen by chance.)
In scientific experiments, researchers are often looking to see if experimental group data are
significantly different from control group data. If the data are significantly different, the
researchers may propose that their treatment (independent variable) within the experimental
group caused the observed difference in the data.
How does a researcher determine if results are significant? By
convention, scientists agree that if there is less than a 5%
chance of getting the observed difference by chance, they
conclude that they have found a statistically significant
difference between the two groups being tested. Therefore,
the total probability value is set at 0.05 (p=0.05).
Two-tailed t-tests are used when the researcher is interested in finding out if the results would
be interesting in either direction (i.e. if the treatment caused a positive or negative effect on the
experimental group). A two-tailed test divides the p-value in half, with 0.025 in each tail. In an
unpaired two-tailed t-test, there is no requirement that the two groups be paired in any way, or
even of equal sizes.
How to conduct an unpaired two-tailed t-test
(1) Define the null hypothesis.
(2) Choose the probability value to be used for establishing significance.
(3) Calculate the t-test statistic. In this handout we’ll call it tcalc. Instead of you calculating it, in
IB Bio the t-test statistic will be calculated for you.
_______________________________________
*The data must be normally distributed with a sample size of at least 10.
(4) Compare tcalc with a table of critical values for t. To use the table you must specify the
degrees of freedom (df) using the formula: df = (n1 + n2) -2, where n1 and n2 are the number of
observations in each data set. For example, if the experimental group has 12 participants, and the
control group has 10 participants, then df = (12+10) – 2 = 20. Then, for a particular df, we look up
the critical value of t (called tcrit) in a t-distribution table.
(5) If tcalc is < tcrit, we accept the null hypothesis. If tcalc is > tcrit, we reject our null hypothesis
Let’s look at an example!
Example # 1:
Two different classes take the same Biology test. Period 2 has 17 students and Period 3 has 15
students. The calculated mean scores, standard deviations, and calculated t-test values are:
Period 2- Ch 51 Test
(out of 62 possible points)
Mean: 53.2 points
Standard Deviation: 4.4
Period 3- Ch 51 Test
(out of 62 possible points)
Mean: 53.9 points
Standard Deviation: 5.5
t-test value: 0.897
1) The null hypothesis: there is no difference between period 2’s test scores and period 3’s test
scores, except that due to random chance.
2) By convention, we will use 5% probability – 2.5% at each end – as our cut-off for significance.
3) The t-test statistic has been calculated for us: 0.897.
4) We determine the degrees of freedom (df) by noting that there are a total of 32 students. We
apply the formula (see previous page) to calculate df.
df
=
(n1 + n2) – 2
=
(17 + 15) – 2
=
32 – 2
=
30
Now we consult a table of t-critical values. Look at 30 degrees of freedom, p = 0.025.
5) The number in the chart, tcrit, is 2.042. Recall, tcalc was 0.897. Since tcalc is < tcrit, we can accept
our null hypothesis – the differences between the period 2 and period 3 test scores are due to
random chance and are not statistically significant.
Now you try one!
Example #2: An experiment was performed to assess whether garlic (in a variety of forms)
inhibits growth in lettuce seedlings. Researchers used 0.5 g of fresh garlic cloves as well as
crushed garlic root, leaf and bulb from garlic cloves that were sprouted for two days. Data
recorded in table 1 are lettuce seedling lengths in millimeters.
Table 1: Lettuce seedling length when exposed to garlic
Trial Number
Fresh garlic
1
2
3
4
5
6
7
8
9
10
4
5
6
4
3
4
3
5
4
3
Lettuce seedling length (mm), when exposed to…
Crushed
Crushed
Crushed
spouted root
spouted leaf
spouted bulb
4
3
3
5
4
4
5
5
5
5
5
4
4
5
5
3
6
4
3
4
5
5
5
4
4
4
4
3
3
5
No garlic
(Control)
20
19
17
20
21
22
19
17
16
20
These raw data are processed. Some of the raw data are shown in Table 2.
Table 2: Processed lettuce seedling length data
Treatment
Mean
seedling
length
Standard
Deviation
(mm)
Calculated t-test
value when
compared to control
Calculated t-test value
when compared to
crushed sprouted bulb
13.9
0.33
(mm)
Fresh garlic
Crushed sprouted root
Crushed sprouted leaf
Crushed sprouted bulb
Control- no treatment
4.1
4.1
4.4
4.3
19.1
0.99
0.88
0.97
0.67
1.91
Example #2 Questions:
1. Is there a significant difference in seedling length in the fresh garlic treatment as compared to the
control?
2. Explain how you were able to make this determination for question 1.
3. Is there a significant difference in seedling length in the fresh garlic treatment as compared to the
crushed sprouted bulb treatment?
4. Explain how you were able to make this determination for question 2.
Name ___________________________________
Per. ____________ Date ____________
t-Test Handout & Statistics- part 2
Now that you have done one example on your own, answer the following questions using your statistics
knowledge.
1)
An experiment was performed using fertilizer on bean plants. Germinated seeds were planted in
sterile soil to which different amounts of commercial fertilizer were added. Heights of the plants were
measured in centimeters 25 days after planting. Data recorded were bean plant heights in centimeters (
0.5 cm).
Table 1: Bean plant height as a function of fertilizer concentration
Trial Number
Treatment
Group B:
Group C:
0.001% fertilizer
0.01% fertilizer
Group A:
no fertilizer
(Height, cm)
1
2
3
4
5
6
7
8
9
10
10
7
8
7
8
10
9
8
7
9
(Height, cm)
(Height, cm)
8
8
7
10
10
7
8
8
9
9
7
10
8
10
8
9
9
8
6
7
Group D:
0.1% fertilizer
(Height, cm)
12
13
15
10
10
15
10
10
14
10
1) Find the Mean and Standard Deviation of each set of data.
(**In Excel: to calculate mean =AVERAGE(data range)
to calculate standard deviation =STDEV(data range).
Treatment
Mean
Standard Deviation
t-Test score when treatment
compared to control
Group A: no fertilizer
Group B: 0.001% fertilizer
Group C: 0.01% fertilizer
Group D: 0.1% fertilizer
2) Compare the means of each group of data.
3) Discuss the variability of the bean plant data using your standard deviation scores.
0.60
2.90
4a) Is there a significant difference in bean plant height between the Group B treatment and the control?
4b) What is the probability that the difference between the two groups is due to chance?
5a) Is there a significant difference in bean plant height between the Group D treatment and the control?
5b) What is the probability that the difference between the two groups is due to chance?
6) When comparing the height of pine trees in two different forests, 32 trees were measured in each
forest. The value of “t” was found to be 2.01. How confident are we that the difference in height of pine
trees in the two forests is a significant difference? (HINT: For your df, round to the closest number on the
chart.)
7) When using the t-test to compare two sets of data, the p-value of the data according to the t-table is
0.05. What does this mean?
8) When using the t-test, what is the level of confidence that is considered a significant difference when
comparing two sets of data?
9) An experiment was performed which measured the size of salmon that spawned in two different streams.
Fifty salmon were sampled for each stream. The value of “t” was found to be 1.29. What is the confidence
level of this particular test?
10) What is the t-test used for?
References: 1) Heinemann Baccalaureate-Standard Level Biology; Pearson Education Limited, 2007 2) http://www.socialresearchmethods.net/kb/stat_t.php