A researcher is going to perform a two-tailed hypothesis test. The test statistic will follow a standard normal distribution.

Answer parts (a) and (b) below. You may use the tool below in the scratch work area to help you. Your work with the tool will not be graded.

(a)The researcher might choose 0.01, 0.05, or 0.10 for the level of significance for the two-tailed test. For each potential choice for the level of significance, find the critical values. Round your answers to three decimal places.
(b)After choosing the level of significance, the researcher takes a sample and finds that the value of the test statistic is ≈ = -1.731. For which of the levels of significance would the null hypothesis be rejected? Choose all that apply.


Let's tackle both parts step by step.

### Part (a) - Finding Critical Values for the Two-Tailed Test

For a two-tailed hypothesis test, we need to find the critical values that correspond to the given levels of significance (\(\alpha\)). The critical values are the z-scores that correspond to the two tails of the standard normal distribution.

The critical values are based on the area in the tails, which is \(\alpha/2\) on each side because it's a two-tailed test.

For each \(\alpha\), we will find the corresponding z-scores:

1. **\(\alpha = 0.01\):**  
   The critical value for \(\alpha = 0.01\) is the z-score that leaves 0.005 in each tail (since 0.01/2 = 0.005).
   We need to find the z-score for the area of \(1 - 0.005 = 0.995\).

2. **\(\alpha = 0.05\):**  
   The critical value for \(\alpha = 0.05\) is the z-score that leaves 0.025 in each tail (since 0.05/2 = 0.025).
   We need to find the z-score for the area of \(1 - 0.025 = 0.975\).

3. **\(\alpha = 0.10\):**  
   The critical value for \(\alpha = 0.10\) is the z-score that leaves 0.05 in each tail (since 0.10/2 = 0.05).
   We need to find the z-score for the area of \(1 - 0.05 = 0.95\).

Using standard normal distribution tables or a calculator, we get the following critical values:

- For \(\alpha = 0.01\), the critical z-scores are approximately \(\pm 2.576\).
- For \(\alpha = 0.05\), the critical z-scores are approximately \(\pm 1.960\).
- For \(\alpha = 0.10\), the critical z-scores are approximately \(\pm 1.645\).

Thus, the critical values for each \(\alpha\) are:
- \(\alpha = 0.01\): \(\pm 2.576\)
- \(\alpha = 0.05\): \(\pm 1.960\)
- \(\alpha = 0.10\): \(\pm 1.645\)

### Part (b) - Checking If the Null Hypothesis Is Rejected

The researcher found the test statistic \(z \approx -1.731\). To determine if the null hypothesis should be rejected, we compare this test statistic to the critical values for each level of significance:

- **For \(\alpha = 0.01\):**  
  The critical values are \(\pm 2.576\). Since \(-1.731\) is not less than \(-2.576\), the test statistic does **not** fall in the rejection region. So, we **do not reject** the null hypothesis at \(\alpha = 0.01\).

- **For \(\alpha = 0.05\):**  
  The critical values are \(\pm 1.960\). Since \(-1.731\) is not less than \(-1.960\), the test statistic does **not** fall in the rejection region. So, we **do not reject** the null hypothesis at \(\alpha = 0.05\).

- **For \(\alpha = 0.10\):**  
  The critical values are \(\pm 1.645\). Since \(-1.731\) is less than \(-1.645\), the test statistic does fall in the rejection region. Therefore, we **reject** the null hypothesis at \(\alpha = 0.10\).

### Conclusion:
- The null hypothesis **is rejected** at \(\alpha = 0.10\).
- The null hypothesis **is not rejected** at \(\alpha = 0.01\) or \(\alpha = 0.05\).

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Would you like more details on any step or concept? Here are five related questions to deepen understanding:

1. What is the difference between one-tailed and two-tailed hypothesis tests?
2. How is the critical value related to the level of significance (\(\alpha\)) in hypothesis testing?
3. Why does lowering the significance level make it harder to reject the null hypothesis?
4. How do you interpret the rejection region in a two-tailed test?
5. What does the test statistic represent in hypothesis testing?

**Tip:** In hypothesis testing, always ensure that the test statistic falls outside the critical values (in the rejection region) before rejecting the null hypothesis.

A researcher is going to perform a two-tailed hypothesis test. The test statistic will follow a standard normal distribution. Answer parts (a) and (b) below. You may use the tool below in the scratch work area to help you. Your work with the tool will not be graded. (a) The researcher might choose 0.01, 0.05, or 0.10 for the level of significance for the two-tailed test. For each potential choice for the level of significance, find the critical values. Round your answers to three decimal places. (b) After choosing the level of significance, the researcher takes a sample and finds that the value of the test statistic is ≈ -1.731. For which of the levels of significance would the null hypothesis be rejected? Choose all that apply.

Learn how to find critical values for a two-tailed hypothesis test using standard normal distribution. Explore solutions for levels of significance (0.01, 0.05, 0.10) and determine when to reject the null hypothesis based on the test statistic value of -1.731. Suitable for college-level statistics students.

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