Find the​ P-value for the indicated hypothesis test with the given standardized test​ statistic, z. Decide whether to reject Upper H 0 for the given level of significance alpha.
​Two-tailed test with test statistic zequalsnegative 1.79 and alphaequals0.03

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.

Learn how to calculate the P-value for a two-tailed hypothesis test with a z-score of -1.79 and a significance level of 0.03. Understand the concept of hypothesis testing, significance levels, and P-values in statistics.

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