Final answer:
To find the critical value with α = 0.02, n = 20, and σ = 2, use a z-distribution table to get the z-value for a cumulative area of 0.98 which is 2.326 for a one-tailed test.
Step-by-step explanation:
To find the critical value for a given significance level (α = 0.02), sample size (n = 20), and known standard deviation (σ = 2), we typically use the z-distribution because the population standard deviation is known. However, because the sample size is less than 30, it's important to know whether we should use the t-distribution or the z-distribution. Assuming a normal distribution or a large enough sample size for the central limit theorem to apply, the z-distribution can be used.
The critical value can be found using a z-table, calculator or statistical software. Since we are given α = 0.02, this implies a one-tailed test. We therefore need to find the z-value that corresponds to the cumulative area of 0.98 (1-α) to find the critical value for the right tail, or 0.02 for the left tail (if required). Using a standard normal distribution table or calculator, we can look up or calculate the z-value that corresponds to the cumulative area. According to the z-table, the critical value for a right-tailed test (α02 = 2.326. In the context of hypothesis testing, if the test statistic is greater than 2.326, we would reject the null hypothesis at the 2% significance level.