To validate the hypothesis that product A is superior to product B in terms of performance, we can set up a hypothesis test. Since we're comparing the means of two independent samples, we will perform a two-sample t-test. Here’s a step-by-step approach to solving the problem:

Step 1: State the Hypotheses

• Null Hypothesis (H₀): The mean life of product A is less than or equal to the mean life of product B. $H_0: \mu_A \leq \mu_B$

• Alternative Hypothesis (H₁): The mean life of product A is greater than the mean life of product B. $H_1: \mu_A > \mu_B$

This is a one-tailed test because we are specifically testing if product A is superior (has a greater mean life) than product B.

Step 2: Define the Parameters

• Sample size for product A (n₁) = 20
• Mean life of product A ($\bar{X}_1$) = 12 months
• Standard deviation for product A (s₁) = 15 days = 15/30 = 0.5 months
• Sample size for product B (n₂) = 20
• Mean life of product B ($\bar{X}_2$) = 10 months
• Standard deviation for product B (s₂) = 10 days = 10/30 = 0.33 months

Step 3: Calculate the Test Statistic

We will use the formula for the two-sample t-test assuming unequal variances (since the standard deviations are different):

$t = \frac{(\bar{X}_1 - \bar{X}_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

Substituting the values:

$t = \frac{(12 - 10)}{\sqrt{\frac{0.5^2}{20} + \frac{0.33^2}{20}}}$

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However, you can manually compute the test statistic using the following values:

$t = \frac{(12 - 10)}{\sqrt{\frac{0.5^2}{20} + \frac{0.33^2}{20}}}$

Once the test statistic is obtained, you can compare it against the critical t-value (using a t-table for 38 degrees of freedom) or calculate the p-value to validate the hypothesis.

Let me know if you want further explanation on how to proceed, or if you have any questions on calculating the test statistic.

Here are 5 related questions for deeper exploration:

1. What is the difference between a one-tailed and two-tailed hypothesis test?
2. How is the degree of freedom calculated for a two-sample t-test?
3. What is the significance level, and how does it relate to the p-value?
4. Can we use a z-test instead of a t-test for this scenario?
5. How do you interpret confidence intervals in hypothesis testing?

Tip: When performing hypothesis testing, always ensure the assumptions of the test (e.g., normality, independence) are met for accurate results.