7.6: Critical Values
A critical value is a definite value obtained from a particular probability distribution at a predecided confidence level (or a predecided significance level) for a given population parameter. The critical value provides demarcation that separates the sample statistics that are likely to occur from the ones that are unlikely to occur based on the given probability distribution and the population parameter to be estimated. The critical value for normal distribution is obtained from the z distribution (z distribution table), commonly known as the z score. For the other non-normal distributions, it can be obtained from the t distribution, F distribution, or Chi-square distribution.
When the sampling distributions of a given population parameter, for instance, population proportion, are normally distributed, the sampling distribution can be converted to the z distribution, and an appropriate z score (the critical z value) is obtained. The common values of getting z scores are at 90%, 95%, and 99% of the confidence level (or at 10%, 5%, or 1% significance level α).
A critical value can be calculated at the right tail, left tail, or both tails of the distribution. The critical value at the right tail is positive, whereas the same at the left tail is negative. For the interval estimate, a critical value is commonly estimated at both tails, generating both positive and negative scores. Thus, the value at half of the significance level α, e.g., α/2, is looked up in the z table to get the critical value at the desired confidence level (for example, z score at 95% confidence level is found by locating 0.9750 in the z table, which generates +1.96 and -1.96). The value of the critical value largely depends on the nature of the hypothesis, the parameter to be estimated, sampling distribution, and in some cases, it can also depend on the sample size. A critical value for interval estimate (i.e., for the given confidence interval) is crucial, without which the confidence limits cannot be calculated.
