To find the p-value for the given test statistic $z = -1.68$ under a two-tailed test, we use the standard normal distribution (Z-distribution).

First, find the cumulative probability corresponding to $z = -1.68$ from the Z-table or using a statistical tool.

Using a Z-table or a calculator:

• The cumulative probability for $z = -1.68$ is approximately 0.0465.

Since it's a two-tailed test, we need to consider both tails:

• For the negative side, the probability is 0.0465.
• For the two-tailed test, we double this value: $2 \times 0.0465 = 0.0930$.

Therefore, the p-value is approximately 0.0930.

Would you like more details or have any questions?

Here are 5 questions you might ask next:

1. How do you use a Z-table to find p-values?
2. What are the steps to perform a Z-test?
3. When should you use a Z-test instead of a t-test?
4. What does it mean if the p-value is greater than the significance level?
5. How can you use software to find p-values more quickly?

Tip: Always check whether your hypothesis test is one-tailed or two-tailed before calculating the p-value, as it affects how you interpret the cumulative probabilities.