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Let's go through each question and provide detailed answers:

### Question 1

**Question:** In this course, the Python methods for hypothesis tests return two-tailed probability values. Suppose a one-tailed alternative hypothesis is used. How can you obtain a one-tailed probability value (P-Value)? Select one.

**Answer:** Divide the result by 2.

**Explanation:** When you have a two-tailed p-value and you want to convert it to a one-tailed p-value, you divide the two-tailed p-value by 2. This is because the two-tailed test considers the extremity of the observed data in both directions, while the one-tailed test considers it in only one direction.

### Question 2

**Question:** The null and alternative hypotheses for a hypothesis test of the difference in two population means are:

- Null Hypothesis: \(\mu_1 = \mu_2\)
- Alternative Hypothesis: \(\mu_1 < \mu_2\)

Notice that the alternative hypothesis is a one-tailed test. Suppose `ttest_ind` method from scipy module is used to perform the test and the output is (3.25, 0.0043). What is the P-value for this hypothesis test? Select one.

**Answer:** 0.0043

**Explanation:** The `ttest_ind` function from the scipy module returns two values: the t-statistic and the two-tailed p-value. Since the question specifies a one-tailed test and the p-value given is already very small (indicating strong evidence against the null hypothesis in one direction), we use the provided p-value directly.

### Question 3

**Question:** What are the inputs to `ttest_ind` method in the scipy module? Select one.

**Answer:** Dataframes of values from each sample and optional equal variance indicator.

**Explanation:** The `ttest_ind` method in the scipy.stats module typically takes two sets of data (arrays or lists) representing the samples from two populations. Additionally, there is an optional parameter `equal_var` which indicates whether the variances of the two populations are assumed to be equal.

### Question 4

**Question:** Which of the following Python methods is used to perform a hypothesis test for the difference in two population proportions? Select one.

**Answer:** `proportions_ztest` from statsmodels module

**Explanation:** The `proportions_ztest` method from the statsmodels module is used to test the hypothesis for the difference in two population proportions.

### Question 5

**Question:** The null and alternative hypotheses for a hypothesis test of the difference in two population proportions are:

- Null Hypothesis: \(p_1 = p_2\)
- Alternative Hypothesis: \(p_1 > p_2\)

Notice that the alternative hypothesis is a one-tailed test. Suppose `proportions_ztest` method from statsmodels is used to perform the test and the output is (1.13, 0.263). What is the P-value for this hypothesis test? Select one.

**Answer:** 0.263

**Explanation:** The `proportions_ztest` function returns the z-statistic and the p-value. Since the test is one-tailed and the output p-value is 0.263, we use this value directly as it represents the probability of observing the test results under the null hypothesis.

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Would you like more details or have any questions? Here are 5 numbered questions you might want to ask next:

1. Can you explain more about the difference between one-tailed and two-tailed tests?
2. What is the significance of the t-statistic in hypothesis testing?
3. How do you interpret the results of a hypothesis test?
4. What are some common Python libraries for performing statistical tests?
5. Can you provide examples of how to use `ttest_ind` and `proportions_ztest` in Python?

**Tip:** Always verify the assumptions behind statistical tests (such as normality, equal variances, etc.) to ensure valid results.

Explore the differences between one-tailed and two-tailed hypothesis tests in statistics. Learn how to interpret results using methods like ttest_ind and proportions_ztest with examples. Ideal for advanced students and professionals.

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