Use the given information to find the​ P-value. Also, use a 0.05 significance level and state the conclusion about the null hypothesis​ (reject the null hypothesis or fail to reject the null​ hypothesis). 

The test statistic in a​ left-tailed test is 

zequals=minus−2.05.

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Part 1

A.

​0.0453; fail to reject the null hypothesis

B.

​0.4798; fail to reject the null hypothesis

C.

​0.0404; reject the null hypothesis

D.

​0.0202; reject the null hypothesis

To solve this, we need to use the **Z-test statistic** to find the **P-value** and make a decision about the null hypothesis at a significance level of \( \alpha = 0.05 \).

1. **Z-score given**: \( z = -2.05 \)
2. **Test type**: Left-tailed

### Step 1: Find the P-value
Since it's a left-tailed test, we are looking for the area to the left of \( z = -2.05 \) in the standard normal distribution.

Using a Z-table or a statistical calculator:

- For \( z = -2.05 \), the corresponding P-value is approximately **0.0202**.

### Step 2: Compare the P-value to the significance level
- **Significance level \( \alpha = 0.05 \)**
- Since the P-value (0.0202) is **less than** the significance level (0.05), we **reject the null hypothesis**.

### Step 3: Conclusion
The correct answer is:
**D. 0.0202; reject the null hypothesis**

Would you like to see more details on how this is done or have further questions? Here are some related questions for practice:

1. What is the interpretation of the P-value in hypothesis testing?
2. How do we use Z-scores in hypothesis tests?
3. What is the difference between left-tailed and right-tailed tests?
4. When do we fail to reject the null hypothesis in hypothesis testing?
5. How do we calculate the P-value for a two-tailed test?
6. What is the role of the significance level \( \alpha \) in hypothesis testing?
7. How does the sample size affect the Z-score and P-value?
8. What are Type I and Type II errors in hypothesis testing?

**Tip**: Always compare the P-value to the significance level to decide whether to reject or fail to reject the null hypothesis.

Learn how to calculate the P-value using a Z-test statistic of z = -2.05 and make a conclusion about the null hypothesis at a significance level of 0.05. Understand the decision-making process in hypothesis testing with detailed steps and explanations.

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