Questions: Using the z table (3 The Standard Normal Distribution Table), find the critical value (or values) for the left-tailed test with α=0.12. Round to two decimal places, and enter the answers separated by a comma if needed.

Using the z table (3 The Standard Normal Distribution Table), find the critical value (or values) for the left-tailed test with α=0.12. Round to two decimal places, and enter the answers separated by a comma if needed.

Transcript text: Using the $z$ table ( 3 The Standard Normal Distribution Table), find the critical value (or values) for the left-tailed test with $\alpha=0.12$. Round to two decimal places, and enter the answers separated by a comma if needed.

Solution

Solution Steps

To find the critical value for a left-tailed test with a given significance level $\alpha=0.12$, we need to determine the z-score that corresponds to the cumulative probability of $1 - \alpha$. This can be done using the inverse of the cumulative distribution function (CDF) for the standard normal distribution.

Step 1: Determine the Cumulative Probability

For a left-tailed test with significance level $ \alpha = 0.12 $, we need to find the cumulative probability corresponding to this significance level. This is given by:

\[ P(Z < z) = \alpha = 0.12 \]

Step 2: Find the Critical Value

To find the critical value $ z $ that corresponds to this cumulative probability, we use the inverse of the cumulative distribution function (CDF) for the standard normal distribution:

\[ z = \Phi^{-1}(0.12) \]

Calculating this gives us:

\[ z \approx -1.1750 \]

Step 3: Round the Critical Value

Rounding the critical value to two decimal places, we have:

\[ z \approx -1.17 \]