A random variable X follows a normal distribution with a mean of \(\mu = 5\) and a standard deviation of \(\sigma = 2\). What is the critical t-value for a 90% confidence interval?
Answer
Step 3: The critical t-value is found by looking up this \(\alpha/2\) value in the t-distribution table. The degrees of freedom for this problem is infinity since we have the standard deviation for the population, not a sample. Looking up 0.05 in the t-table, we get a critical value of approximately 1.645.
Step 1 ：Step 1: First, we need to find the value of \(\alpha\). Since the confidence level is 90%, then \(\alpha = 1 - 0.90 = 0.10\).
Step 2 ：Step 2: Because we are working with a two-tailed test, we need to divide \(\alpha\) by 2. Therefore, \(\alpha/2 = 0.10/2 = 0.05\).
Step 3 ：Step 3: The critical t-value is found by looking up this \(\alpha/2\) value in the t-distribution table. The degrees of freedom for this problem is infinity since we have the standard deviation for the population, not a sample. Looking up 0.05 in the t-table, we get a critical value of approximately 1.645.
