Validate the hypothesis that product A is superior to product B in terms of performance. A sample of 20 items of product A is having mean life of 12 months with standard deviation 15 days where as product B is having mean life of 10 months with standard deviation 10 days. Use p – value and validate the hypothesis.

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:

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

Let me compute that.It seems like I can’t do more advanced data analysis right now. Please try again later.

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.

Validate the hypothesis that product A is superior to product B in terms of performance. A sample of 20 items of product A is having mean life of 12 months with standard deviation 15 days whereas product B is having mean life of 10 months with standard deviation 10 days. Use p-value and validate the hypothesis.

Learn how to validate the hypothesis that product A is superior to product B in terms of performance using a two-sample t-test. Compare the mean life of product A (12 months) and product B (10 months) with sample sizes of 20, using their respective standard deviations. This solution covers hypothesis testing, p-values, and the t-distribution theorem.

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