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.