To find the $P$-value for the given two-tailed hypothesis test with a standardized test statistic $z = -1.79$ and a significance level $\alpha = 0.03$, follow these steps:

1. Understand the Test:

   • Two-tailed test: The alternative hypothesis ($H_1$) states that the parameter is not equal to the null hypothesis value ($H_0$).
   • Test statistic: $z = -1.79$
   • Significance level: $\alpha = 0.03$

2. Find the $P$-value:

   • The $P$-value for a two-tailed test is the probability that the test statistic is at least as extreme as the observed value in either direction (both tails).
   • First, find the $P$-value for the left tail using the $z$-score.

3. Calculate the Left Tail Probability:

   • Using the standard normal distribution table (or a calculator), find the probability corresponding to $z = -1.79$.
   • This probability is the area to the left of $z = -1.79$.

$P(Z < -1.79) \approx 0.0367$

4. Calculate the $P$-value for the Two-Tailed Test:
   • Since the test is two-tailed, multiply the left tail probability by 2.

$P \text{-value} = 2 \times P(Z < -1.79) \approx 2 \times 0.0367 = 0.0734$

5. Compare the $P$-value with $\alpha$:
   • $\alpha = 0.03$

Since the $P$-value (0.0734) is greater than the significance level ($\alpha = 0.03$), we fail to reject the null hypothesis $H_0$.

Conclusion:

• The $P$-value is approximately 0.0734.
• Since the $P$-value is greater than $\alpha = 0.03$, we do not have sufficient evidence to reject the null hypothesis at the 0.03 significance level.

Would you like more details or have any questions?

Related Questions

1. What is the significance level in hypothesis testing?
2. How do you interpret a $P$-value in hypothesis testing?
3. What does it mean to reject the null hypothesis?
4. How do you find the $z$-score for a given data set?
5. What is the difference between a one-tailed and a two-tailed test?
6. Can you explain the standard normal distribution?
7. How do you use the standard normal distribution table?
8. What are Type I and Type II errors in hypothesis testing?

Tip: Always ensure you understand whether the test is one-tailed or two-tailed as this affects the calculation of the $P$-value significantly.