Final answer:
To calculate the minimum sample size using the critical value, apply a sample size formula that considers a specified margin of error and confidence level. Exact sample size cannot be determined without the actual critical value or standard deviation. Larger sample sizes yield more accurate estimates with higher confidence.
Step-by-step explanation:
To determine the minimum sample size needed when using a critical value for estimating a population proportion with a certain level of confidence, the student should use a sample size formula developed for proportions. Typically, this involves setting up the equation with a given margin of error, the critical value for a chosen confidence level (found from a Z-table or T-table), and the estimated proportion. It is clear that the goal is to find a sample size that meets the requirements for margin of error at a specific confidence level.
In the question given, we are told that an estimate of the total population is 400 individuals, and the desired confidence level is 90% with a margin of error of 0.05. Using the provided information, we are looking for a sample size for which 90% of the confidence intervals would contain the true population mean. According to the sampling distribution principles, especially when dealing with proportions, the requirement is that the number of successes and failures should both be at least five.
Given that larger samples provide more accurate estimates with higher confidence, if a student asks for the minimum sample size with a critical value and certain parameters, an appropriate answer would reference the formula for calculating sample size for estimates and provide a numerical answer. Without the exact critical value or standard deviation given in the question, however, it's impossible to provide an accurate numerical value to the question. Hence, the student should be advised to obtain these values first or use typical values for a 90% confidence level to compute the sample size.