What are the considerations in determining the sample size and sampling technique of the study?
3. Production Process Characterization
\( \mbox{Var}(\hat{p}) \approx \frac{\sigma^2}{n} \) Show
with \(\hat{p}\) denoting the parameter we are trying to estimate. This means that if the variability of the population is large, then we must take many samples. Conversely, a small population variance means we don't have to take as many samples. Practicality Of course the sample size you select must make sense. This is where the trade-offs usually occur. We want to take enough observations to obtain reasonably precise estimates of the parameters of interest but we also want to do this within a practical resource budget. The important thing is to quantify the risks associated with the chosen sample size. Sample size determination In summary, the steps involved in estimating a sample size are:
\( Pr(|\hat{p} - P| \ge \delta) = \alpha \) where
\( n = z_{\alpha}^{2}(\frac{pq}{\delta^2}) \) where z is the ordinate on the Normal curve corresponding to α. Example Let's say we have a new process we want to try. We plan to run the new process and sample the output for yield (good/bad). Our current process has been yielding 65% (p=.65, q=.35). We decide that we want the estimate of the new process yield to be accurate to within δ = .10 at 95% confidence (α = .05, zα = -2). Using the formula above we get a sample size estimate of n=91. Thus, if we draw 91 random parts from the output of the new process and estimate the yield, then we are 95% sure the yield estimate is within .10 of the true process yield. Estimating location: relative error If we are sampling continuous normally distributed variables, quite often we are concerned about the relative error of our estimates rather than the absolute error. The probability statement connecting the desired precision to the sample size is given by:\( Pr\left( \left\|\frac{\hat{y} - \mu}{\mu}\right\| \ge \delta) \right) = \alpha \) where μ is the (unknown) population mean and \(\bar{y}\) is the sample mean. Again, using the normality assumptions we obtain the estimated sample size to be: \( n \approx \frac{z_{\alpha}^{2}\sigma^{2}}{\delta^{2}\mu^{2}} \) with σ2 denoting the population variance. Estimating location: absolute error If instead of relative error, we wish to use absolute error, the equation for sample size looks alot like the one for the case of proportions:\( n \approx z_{\alpha}^{2}\left( \frac{\sigma^{2}}{\delta^{2}} \right) \) where σ is the population standard deviation (but in practice is usually replaced by an engineering guesstimate). Example Suppose we want to sample a stable process that deposits a 500 Angstrom film on a semiconductor wafer in order to determine the process mean so that we can set up a control chart on the process. We want to estimate the mean within 10 Angstroms (δ = 10) of the true mean with 95% confidence (α = .05, zα = -2). Our initial guess regarding the variation in the process is that one standard deviation is about 20 Angstroms. This gives a sample size estimate of n = 16. Thus, if we take at least 16 samples from this process and estimate the mean film thickness, we can be 95% sure that the estimate is within 10 angstroms of the true mean value.What are the considerations in determining the sample size?When choosing a sample size, we must consider the following issues:. What population parameters we want to estimate.. Cost of sampling (importance of information). How much is already known.. Spread (variability) of the population.. Practicality: how hard is it to collect data.. How precise we want the final estimates to be.. What are practical considerations in sampling and sample size?The risks around using a sample to make conclusions about a population are only one of three considerations when determining the sample size for an experiment. The sampling risk, the population's variance, and the precision or amount of change we wish to detect all impact the calculation of sample size.
What are the considerations that determine the nature of sample size in quantitative research?They are significance level, power and effect size.
What are three key considerations that will influence your sample size?3 Key Factors to Consider When Determining the Right Sample Size. Know how variable the population is that you want to measure. ... . Know how precise the population statistics need to be. ... . Know exactly how confident you must be in the results.. |