The image shows a hypothesis test with a two-tailed alternative hypothesis $H_1: \mu \neq 6$.

For part (a), the test statistic is $z = 1.21$, and the calculated $P$-value is 0.2263.

Now, in part (b), the test statistic is $z = -0.99$. For a two-tailed test, we need to find the $P$-value corresponding to $|z| = 0.99$.

1. Calculate the P-value for $z = -0.99$:
   • Using standard normal distribution tables or a calculator:
     • For $z = 0.99$, the one-tailed area is approximately 0.1611.
     • For a two-tailed test, the $P$-value is $2 \times 0.1611 = 0.3222$.

So, the $P$-value is approximately 0.3222.

Would you like a detailed explanation of the interpretation of the $P$-value, or have any other questions?

Here are five questions to deepen understanding:

1. Why is the $P$-value multiplied by 2 in a two-tailed test?
2. How do we interpret a $P$-value of 0.3222 in the context of hypothesis testing?
3. What does the $z$-score represent in this type of hypothesis test?
4. How would the $P$-value change if this were a one-tailed test?
5. When is it appropriate to use a two-tailed vs. a one-tailed test?

Tip: In hypothesis testing, a smaller $P$-value (e.g., less than 0.05) suggests stronger evidence against the null hypothesis.