Part 1:
1. Calculate the sample size needed given these factors:
The factors given for calculating the sample size are as follows:
one-tailed t-test with two independent groups of equal size
small effect size = 0.2
alpha = 0.05
beta = 0.2
Power = 1 – beta = 1 – 0.2 = 0.8
Using the priori function:
Using the priori function on GPower, the sample size based on the factors above was determined. The results of the analysis are as shown below.
Results:
t tests - Means: Difference between two independent means (two groups)
Analysis: A priori: Compute required sample size
Input: Tail(s) = One
Effect size d = 0.2
α err prob = 0.05
Power (1-β err prob) = 0.8
Allocation ratio N2/N1 = 1
Output: Noncentrality parameter δ = 2.4899799
Critical t = 1.6473230
Df = 618
Sample size group 1 = 310
Sample size group 2 = 310
Total sample size = 620
Actual power = 0.8002178
Using the compromise function:
Results:
t tests - Means: Difference between two independent means (two groups)
Analysis: Compromise: Compute implied α & power
Input: Tail(s) = One
Effect size d = 0.2
β/α ratio = 4
Sample size group 1 = 155
Sample size group 2 = 155
Output: Noncentrality parameter δ = 1.7606817
Critical t = 1.3655189
Df = 308
α err prob = 0.0865432
β err prob = 0.3461729
Power (1-β err prob) = 0.6538271
The results of the analysis carried out indicate that for half the sample size and a β/α ratio equal to four, α err prob is equal to 0.086 while the β err prob equal to 0.3461. This value of β err prob provides a power of 0.653 for the study. This indicates that as the sample size decreases the β err prob increases and in turn decreases the power of the study.
Rationale for the β/α ratio:
The β/α ratio used in the power analysis above equals to four. The main reason for selecting this ratio is to ensure a high statistical power. This is achieved by ensuring the right balance between Type I and Type II errors. In turn, this ration ensures that the study is neither underpowered nor overpowered (Faul, Erdfelder, Lang, & Buchner, 2007).
Reasons to use a small sample:
The study is worth doing using a smaller sample size due to a number of reasons. Acheson (2010) points out that selecting the right sample size is critical in any research study. According to Nayak (2010), a small sample is desirable in answering some research questions due to its advantages. The main advantage the article suggests is that large samples can be a waste of resources. The author points out that in some studies smaller samples sufficiently answer the research questions. Therefore, a researcher does not need a large sample. Furthermore, larger samples are resource and time intensive for a researcher to collect data sufficiently. This makes the use of a smaller sample size desirable because it is cheaper and requires fewer resources to collect data. In addition, in research that involves human subjects a large sample may be considered unethical (Nayak, 2010).
Part 2:
2. Calculate the sample size needed given these factors:
The factors given for calculating the sample size are as follows:
ANOVA (fixed effects, omnibus, one-way)
Small effect size
alpha =.05
beta = .2
3 groups
Using the priori function:
Using the priori function on GPower, the sample size based on the factors above was determined. The results of the analysis are as shown below.
Results:
F tests - ANOVA: Fixed effects, omnibus, one-way
Analysis: A priori: Compute required sample size
Input: Effect size f = 0.1
α err prob = 0.05
Power (1-β err prob) = 0.8
Output: Noncentrality parameter λ = 9.6900000
Critical F = 3.0050418
Numerator df = 2
Denominator df = 966
Total sample size = 969
Actual power = 0.8011010
Using the compromise function:
Results:
Analysis: Compromise: Compute implied α & power
Input: Effect size f = 0.1
β/α ratio = 4
Total sample size = 486
Output: Noncentrality parameter λ = 4.8600000
Critical F = 2.3398285
Numerator df = 2
Denominator df = 483
α err prob = 0.0974353
β err prob = 0.3897414
Power (1-β err prob) = 0.6102586
Rationale for β/α ratio:
The β/α ratio utilized equals to four. The main reason for choosing a high ratio is to ensure that the value of β is greater than alpha, hence leading to a higher power (1-β err prob).
Reasons for using a smaller sample:
According to Thomas & Juanes (1996), large samples are also undesirable since they may lead to significant results for any statistical test that may not be biologically significant. This same view is held by Houser (2007) who points out that large samples may lead to statistically significant results that may not be clinically significant. Furthermore, Houser (2007) also argues that large samples are costly and increase the duration needed to complete a study. Lastly, for studies with large effects, a small sample can easily detect the phenomenon under study (Houser, 2007). Based on the reasons discussed above it is evident that a smaller sample size is desirable in some studies.
Part 3:
Research question: Do students enrolled in on-campus classes learn more as compared to students enrolled for online classes?
Study design 1: Observational study
For the observational study, the researcher does not manipulate the subjects under study (Trochim & Donnelly, 2008). However, the researcher will observe both groups of students throughout the course of a semester. During the period, the researcher will collect data relating the student’s performance. An ANOVA (fixed effects, omnibus, one-way) will find use in comparing the results from the two groups.
The factors selected are:
A medium effect size since the researcher hypothesizes that a medium effect exists in the topic under study.
Alpha is equal to 0.05 providing a 5% probability of type I error
Beta is equal to 0.05 providing a 5% probability of type II errors
2 groups
Results:
Analysis: A priori: Compute required sample size
Input: Effect size f = 0.25
α err prob = 0.05
Power (1-β err prob) = 0.95
Output: Noncentrality parameter λ = 13.1250000
Critical F = 3.8865546
Numerator df = 1
Denominator df = 208
Total sample size = 210
Actual power = 0.9501287
Study design 2: Experimental design
An experimental design will be employed in determining whether students enrolled in on-campus college courses learn more than students enrolled in e-classes. For the experiment, the control group is on campus students who will receive regular course instruction. On the other hand, the treatment group is the e-classes students who will receive special course instruction over the internet. A One tailed T-test to test the difference between groups will find use in determining whether there is a significant difference in the performance of the two groups of students.
The factors selected are:
A medium effect size since the researcher hypothesizes that a medium effect exists in the topic under study.
Alpha is equal to 0.05 providing a 5% probability of type I error
Beta is equal to 0.2 providing a 20% probability of type II errors
Power is equal to 0.8 since it is equal to 1 – beta = 1 – 0.2 = 0.8
Results:
Input: Tail(s) = One
Effect size d = 0.5
α err prob = 0.05
Power (1-β err prob) = 0.8
Allocation ratio N2/N1 = 1
Output: Noncentrality parameter δ = 2.5248762
Critical t = 1.6602343
Df = 100
Sample size group 1 = 51
Sample size group 2 = 51
Total sample size = 102
Actual power = 0.8058986
References:
Acheson, A. (2010). Sample size. http://sk.sagepub.com.proxy1.ncu.edu/reference/researchdesign/n396.xml
Faul, F., Erdfelder, E., Lang, A. G., & Buchner, A. (2007). G* Power 3: A flexible statistical power analysis program for the social, behavioral, and biomedical sciences. Behavior research methods, 39(2), 175-191.
Houser, J. (2007). How many are enough? Statistical power analysis and sample size estimation in clinical research. J Clin Res Best Pract, 3(3), 1-5.
Mayr, S., Erdfelder, E., Buchner, A., & Faul, F. (2007). A short tutorial of GPower. Tutorials in Quantitative Methods for Psychology, 3(2), 51-59.
Nayak, B. K. (2010). Understanding the relevance of sample size calculation. Indian journal of ophthalmology, 58(6), 469.
Thomas, L., & Juanes, F. (1996). The importance of statistical power analysis: an example fromAnimal Behaviour. Animal Behaviour, 52(4), 856-859.
Trochim, W. M. K., & Donnelly, J. P. (2008). Research methods knowledge base. Mason, Ohio: Atomic Dog/Cengage Learning.
