Provide an example of a hypothetical study that would use a one-way between-subjects ANOVA. Be sure to describe the different groups, conditions, and hypotheses involved. What would the null be and does successful rejection indicate a mean significantly different from every other mean? How would you decide on the necessary sample size (one which provides adequate statistical power)?

The results of data analysis of research conducted in a sales organization that operates in 50 different cities of the country and employees a total sales force about 500. The number of sales persons sampled for the study was 150.

The sales persons were divided in 5 levels of education. Based on education, groups were formed and sales were measured:

The hypothesis can be defined as:

H0: the sales do not depend on the education level of sales person

      (the mean sales for all groups based on education will remain same)

Vs

H1: the sales depend on level of education of sales person.

       (the mean sales for all groups based on education will be different)

The above table shows significance p - Value = 0.01 <0.05 

Conclusion:

At 5% level of significance reject null hypothesis. This means that at 5% level of significance we are unable to accept the average sales based on education is equal.

This further means that if we want to test individually for different levels of education, we need to conduct separate t-tests for each group data.

Step-by-step explanation

How would you decide on the necessary sample size (one which provides adequate statistical power)?

Practically speaking, the higher the sample size the higher is the power of the test.

Power is greatest when the alfa level, difference between group means, and sample size are large and the variability among subjects is small. Although greater power is achieved with a larger, rather than a smaller, a level, recall that the likelihood of making a type I error is also increased when alfa is set at a high level. Assuming researchers do not want to risk contributing false findings to the literature, they will seldom set the alfa level above 0.05. Instead, they will attempt to insure a large mean difference by selecting 2 treatment conditions that are likely to produce substantial differences, minimize variability among subjects by narrowly defining the population of interest, carefully controlling the measurement setting, and using a large sample size. 

EFFECT SIZE 

The difference between group means and variability are often combined into one number that is called an effect size. 

POWER ANALYSIS 

Power analysis is a technique based upon the interrelationships among power, alfa level, effect size, and sample size. The relationships among these factors are such that values for any 3 of the factors can be used to estimate the last. 

Estimating Sample Size 

Use power analysis to estimate the sample size (N) required to yield statistically significant results from a study. The value for the alfa level is generally set at 0.01 or 0.05. 

Given the data , we could calculate a standardized index of effect size using the formula 

d = (M1 - M2)/s, where M1 is the mean of group 1, M2 is the mean of group 2, and s is the pooled standard deviation of both groups. 

According to Cohen, values of 0.80, 0.50, and 0.20 are considered large, medium, and small effect sizes, respectively. 

Note, though, that relying on such crude benchmarks should be avoided if information specific to your research area is available. If we expected very little difference in ROM between the 2 exercise groups, we would be wise to select a small effect size. Conversely, if we expected a large difference in ROM between the 2 exercise groups, we would select the large effect size. 

Next, we would need to specify the level of power considered acceptable for our study. 

The power can be defined as 1 - P where P represents the probability of making a type 2 error (ie, a missed finding). When choosing a power level, the researchers must decide how willing they are to make a type 2 error. By convention, the minimum power value typically used is 0.80.