Determining the Sample Size

Doing research costs… Power of a hypothesis test generally is an increasing function of sample size. Margin of error is generally a decreasing function of sample size. Cost of research is generally an increasing function of sample size Who is paying the bills?

Before research project begins…  Formulation of hypotheses.  Choice of significance/confidence levels.  What size of effect are we looking for?  What power do we want?  How many observations do we need?  Can we afford it?

Research Setting You are a statistician working for a manufacturer. Process engineer is investigating the mean amount of time required to complete an assembly task. He wishes to show that the mean assembly time is less than 30 seconds. Past experience with similar studies leads him to believe that the assembly times will be approximately normally distributed, and that the sample range for 1000 observations will be approximately 9 seconds.

 Formulation of Hypotheses: In this case, H a :  < 30 sec., so that H 0 :   30 sec.  Choice of significance/confidence levels: We weigh the possible consequences of making either a Type I error or a Type II error, and decide that we want  = We also want to obtain a 95% confidence interval estimate of the mean assembly time.  The engineer tells us that we want to be able to detect a mean difference of 0.5 sec., i.e., if the mean assembly time is less than 29.5 sec., we should be able to detect it.

 We want to be able to detect this size of effect with probability We also want to be able to estimate the mean assembly time with an interval width of 0.8 sec.

What size sample do we need? Our test statistic is. Under H 0, this statistic has a t(n-1) distribution. What is The distribution under H a ? It is noncentral with noncentrality parameter.

Power analysis If T n-1,  is a random variable having the above non- central t distribution, then we see that the power of the test is the probability that this r.v. will be found to be less than the critical value of the test. Thus the power depends on the true value of , the sample size n, and the true population standard deviation . We know what size of effect we want to be able to detect, but we need to know something about .

“Guess-timating”  If the range of values of assembly times for 1000 observations is 9 sec., and if assembly time is a normally distributed r.v., then we can “guess-timate” that.

Sample Size for Hypothesis Test We then want to find n so that. Can we use SAS to do this calculation? If we run the following program using a range of possible sample sizes, we will be able to solve our problem.

data one; input n; cp = tinv(0.05,n-1); delta = 0.5/(1.5/sqrt(n)); power = probt(cp,n-1,-delta); put cp power; cards; 50 ; run;

Sample Size for Estimation The form of the confidence interval is. We want the margin of error to be 0.4 sec. We then want a value of n to satisfy the following inequality:

. The following SAS program, run for a range of values of n, will help us to solve our problem.

data one; input n; cp = -tinv(0.025,n-1); numg = gamma(n/2); deng = gamma((n-1)/2); width = (2*cp*sqrt(2)*numg*1.5)/(sqrt(n*(n-1))*deng); put n cp width; cards; 50 ; run;

Sample Size We had two criteria for the sample size; one based on the power of the test, the other based on the margin of error of estimation. We choose the larger of the two values of n to be sure that we achieve both the desired power and the desired margin of error.

To obtain our sample sizes, we had to make an assumption about the value of . If our “guess-timate” for  was too small, then our power would be less that the desired value and our margin of error would be larger than the desired value. If our “guess-timate” was too large, then we might be wasting resources with a sample size that would be larger than necessary.