Step 4: Make a Decision and Report the Results
I. Practicing With the t-Table
For each of the obtained t-values given below, determine the t-critical value and decide if H0 should be rejected. For the one-tailed tests, assume that the researcher is looking for score increases.
df α level One or two- tailed test
tobt tcrit Reject H0 Y/N
A. 11 .01 two-tailed 4.32 ________ ________
B. 80 .05 two-tailed 1.04 ________ ________
C. 24 .05 one-tailed 2.18 ________ ________
D. 24 .01 one-tailed 2.18 _______ ________
E. 42 .05 one-tailed 1.87 _______ ________
F. 42 .05 two-tailed 1.87 _______ ________
G. 19 .01 two-tailed 3.06 _______ ________
H. 19 .001 two-tailed 3.06 _______ ________
Notice that, given the same degrees of freedom and alpha level, it is more difficult to reject the null hypothesis using a two-tailed test than a one-tailed test. In addition, observe that the null hypothesis is more difficult to reject as we move from a .05 to a .01 alpha level, and it is even more difficult to reject at α = .001.
Tip!
ONE-SAMPLE t TEST
The one-sample t test is a test of a hypothesis about a population mean (μ) when the popula- tion standard deviation (σ) is not known. This test is used when researchers want to know (1) if a sample is representative of a population and/or (2) if a particular treatment or condi- tion has a significant effect. We will use the same four-step procedure that we used for the z-test, beginning with a research question.
Sample Research Question (Two-Tailed Test)
The population mean on a standardized test of critical thinking is μ = 53. A group of faculty members at a small community college underwent a 10-week training program to learn tech- niques designed to help students develop their critical thinking skills. After the training, the new techniques were implemented in the classrooms. The mean critical thinking score for a sample of n = 87 students exposed to the new techniques was M = 55 with SS = 6013. Do the results suggest that the training program had a significant effect? Use a two-tailed test and α = .05.
Step 1: Formulate Hypotheses H0: μ = 53
H1: μ ≠ 53
The null hypothesis asserts that the critical thinking scores in the population, after implemen- tation of the new techniques, would still be 53. We are using the non-directional alternative hypothesis that the population mean would be some value other than the one specified in H0. Step 2: Indicate the Alpha Level and Determine Critical Values
α = .05 df = 86 tcrit = ±2.000
The actual df is 86 (i.e., n−1). However, this value is not shown in the t-distribution table.
Thus, we will use the next-lowest df, which is 60, for which the chart indicates a value of 2.00.
Step 3: Calculate Relevant Statistics
Because σ is not known, we will need to estimate it from our sample data. For this problem, rather than calculating the standard deviation from the raw scores, you are given the value for SS (sum of squares). We will use this value for calculating the estimated population standard deviation(s):
s SS n
1
6013 87 1 8 36.
Before we can determine our obtained t-value, we first need to calculate the standard error using the estimated standard deviation above:
S s
M = n =8 36= 87 90
. .
Finally, we can calculate the t-statistic:
t M
SM
obt 55 53
90 2 22
. .
The t-distribution for this example is below:
The tcrit values are ±2.000. To reject H0, we need a sample mean with an obtained t-value beyond 2.000 in either direction. Our obtained t-value was +2.22, which falls in the rejection region. Thus, we reject the null hypothesis.
Proper Format
The format for reporting the results of a one-sample t test is as follows:
Students taught by the faculty who participated in the training program scored signifi- cantly higher on the critical-thinking assessment than did the general population. Reject H0, t(86) = +2.22, p < .05.
Note that
• our obtained probability is less than .05. In other words, if the null hypothesis were true (i.e., if μ = 53), our obtained sample mean would have occurred by chance less than 5% of the time. This unlikely event leads us to reject the null hypothesis and to conclude instead that the critical-thinking training program significantly influenced students’ scores.
• the t-value reported is the obtained t, not the critical t.
• the number in parentheses after t is the actual degrees of freedom, not the lower df listed in the chart.
EFFECT SIZE FOR A ONE-SAMPLE t TEST
Because our results were significant, we need to ask how significant they were. Here again, we will use Cohen’s d to measure the size of the effect, as we did in the last chapter for the z-test.
However, the formula as applied to a one-sample t test will change slightly, as shown below, because for the t test we used an estimated population standard deviation (s) in lieu of an actual population standard deviation (σ).
General formula: For our current problem:
d M
s
d
55 53 8 36 24
. .
The critical-thinking scores of the students showed an improvement of about one-fourth of a standard deviation (.24) above the mean of the general population.
As a reminder, the guidelines recommended by Cohen in the last chapter are repeated below:
• d = .20 to .49 – Small effect
• d = .50 to .79 – Moderate effect
• d = .80 and above – Large effect
Thus, the results of our t test can be interpreted as exhibiting a small effect size.
Another Sample Research Question (One-Tailed Test)
Remember, if a problem indicates that a researcher is looking for score increases or decreases, then assume a one-tailed test. If the researcher is simply looking for an effect with no indica- tion of direction, then assume a two-tailed test.
A well-known sandwich chain puts 9 grams of protein on its sandwiches. A customer complained to the home office that a particular outlet was putting “hardly any meat” on its sandwiches. A random sample of n = 16 sandwiches from the sandwich shop in question were weighed. The results showed a M = 7.9 with s = 2.1. Did the shop put significantly less protein on its sandwiches? Test at α = .05.
Step 1: Formulate Hypotheses H0: μ ≥ 53
H1: μ < 53
Step 2: Indicate the Alpha Level and Determine Critical Values α = .05
df = 15 tcrit = – 2.131
Step 3: Calculate Relevant Statistics
S s
M = n = 2 1 = 16. 53
.
t M
SM
obt 7 9 9 53 2 08 .
. .
Step 4: Make a Decision and Report the Results
The sandwich shop did not put a significantly less amount of protein on its sandwiches. Fail to reject H0, t(15) = −2.08, p > .05.
ASSUMPTIONS
The same assumptions that apply to the z-test also apply to the t test, with one exception. The applicable assumptions include the following:
• Independent and random selection of participants.
• The dependent variable is normally distributed in the population of interest.
• The dependent variable can be measured on an interval or ratio scale.
The only assumption that pertains to the z-test that does not pertain to the t test is that the population standard deviation (σ) is known. Because we are estimating the population standard deviation for the t test, this requirement does not apply.
Your Turn!