Download What is statistics

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, NP, 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 instance, the Chi-Square Test, the Pearson Correlation Test, and… a
t-test. 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, the 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.
Two-tailed t-tests are used when the researcher wants to finding out if the results would be
interesting in either direction (i.e. if the treatment had a positive or negative effect on the
experimental group). And 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) Determine the t-test statistic. In this handout we’ll call it tcalc. It's time-consuming to
calculate by hand so in this handout we'll calculate it for you.
3) 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 an experimental group has 12
participants, and a control group has 10 participants, then df = (12+10) – 2 = 20.
4) Look up the critical value of t (called tcrit) in a t-distribution table for the particular df at P =
0.05.
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!
Two different classes take the same Biology test. Period 1 has 17 students and Period 3 has 15
students. The calculated mean scores, standard deviations, and calculated t-test values are:
_______________________________________
*The data should be normally distributed with a sample size of at least 10.
Table 1. Period 1 & 3 Test Results, Out of 62 Possible Points
Period 1
Period 3
Mean: 53.2 points
Mean: 53.9 points
Standard Deviation: 4.4
Standard Deviation: 5.5
t-test value (calculated for you): 0.897
1) The null hypothesis: there is no difference between period 1’s and period 3’s test scores,
except that due to random chance.
2) The t-test statistic has been calculated for us in this example: 0.897.
3) We determine the degrees of freedom (df) by applying the formula given above.
df
=
(n1 + n2) – 2
=
(17 + 15) – 2
=
32 – 2
=
30
4) Now we consult a table of t-critical values.
Looking in the row for 30 degrees of freedom,
tcrit is 2.04 at P = 0.05.
5) Recall, tcalc was 0.897, which is < tcrit (2.04).
Thus we accept our null hypothesis: the
difference between the period 1 and period 3
mean test scores is due to random chance and
is therefore not statistically significant.
Table 2. t-distribution
table
IHS / IB Biology
Just My Cup of Tea: Student Worksheet
1. 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 are shown in table 1.
Table 1: Lettuce seedling length when exposed to 0.5 g of different forms of garlic
Trial Number
1
2
3
4
5
6
7
8
9
10
Mean seedling
length (mm)
Standard dev'n
(mm)
a.
Fresh garlic
4
5
6
4
3
4
3
5
4
3
4.1
0.99
Lettuce seedling length (mm), when exposed to…
Crushed
Crushed
Crushed
sprouted garlic
sprouted garlic
sprouted garlic
root
leaves
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
4.1
4.4
4.3
0.88
0.97
0.68
No garlic
(Control)
20
19
17
20
21
22
19
17
16
20
19.1
1.91
When comparing the mean seedling length of lettuce seeds exposed to fresh garlic as compared to the
control, tcalc = 13.9. Is this difference statistically significant?
b. Explain how you were able to make this determination for (a), including how you figured out tcrit.
c.
When comparing the mean seedling length of lettuce seeds exposed to fresh garlic as compared to the
crushed sprouted garlic, tcalc = 0.33. Is this difference statistically significant?
d. Explain how you were able to make this determination for (c), including how you figured out tcrit.
2) When comparing the height of pine trees in two different forests, 32 trees were measured in each
forest. The value of “tcalc” was found to be 2.01. Is the difference in average height of the pine trees in the
two forests significant? Explain! (HINT: For your df, round to the closest number on the chart.)
3) An experiment was performed that measured the size of salmon that spawned in two different streams.
48 salmon were sampled for one stream and 52 for the other stream. The value of “tcalc” was found to be
1.29. Is the average size of the salmon in each stream significantly different? Explain!