Question 1192645
Certainly, let's break down the assignments and guide you through the solutions.

**Assignment 3**

**1. Constructing X-bar and R-charts**

* **Calculate the mean (X̄) and range (R) for each group:**
    * For each group, calculate the mean of the 5 thickness measurements.
    * For each group, calculate the range (difference between the highest and lowest values).

* **Calculate the average of means (X̄̄) and average of ranges (R̄):**
    * Find the average of the means calculated for each group.
    * Find the average of the ranges calculated for each group.

* **Determine control limits:**
    * **X-bar chart:**
        * Upper Control Limit (UCL) = X̄̄ + A2 * R̄ 
        * Lower Control Limit (LCL) = X̄̄ - A2 * R̄
    * **R-chart:**
        * Upper Control Limit (UCL) = D4 * R̄
        * Lower Control Limit (LCL) = D3 * R̄

    * A2, D3, and D4 are constants obtained from control chart constants tables based on the sample size (n = 5 in this case).

* **Plot the data:**
    * Plot the group means on the X-bar chart.
    * Plot the group ranges on the R-chart.

* **Analyze the charts:**
    * Check if any points fall outside the control limits. 
    * Look for any patterns or trends in the data.

**2. Testing for a Difference Between Machines P and Q**

* **Calculate the means and standard deviations for each machine.**
* **Perform a t-test:**
    * **Hypothesis:**
        * Null hypothesis (H0): The means of the two machines are equal.
        * Alternative hypothesis (H1): The means of the two machines are not equal.
    * **Calculate the t-statistic:** 
        * Use the formula for the two-sample t-test, taking into account the possibility of unequal variances.
    * **Determine the degrees of freedom.**
    * **Find the critical t-value** based on the degrees of freedom and the chosen significance level (e.g., 0.05 for a 95% confidence level).
    * **Compare the calculated t-statistic to the critical t-value.**
    * **Make a decision:** 
        * If the calculated t-statistic is greater than the critical t-value or less than the negative of the critical t-value, reject the null hypothesis. 
        * Otherwise, fail to reject the null hypothesis.

**C.A.T 2**

**1. Finding the Probability of a Bulb Lasting More Than 648 Hours**

* **Use the given probabilities:**
    * P(lifetime > 682 hours) = 0.9788
    * P(lifetime > 703 hours) = 0.0051

* **Determine the probability of a bulb lasting between 682 and 703 hours:**
    * P(682 < lifetime <= 703) = P(lifetime > 682 hours) - P(lifetime > 703 hours) 

* **Assume a continuous distribution (e.g., exponential or Weibull - this may need to be specified in the problem context).**
* **Use the cumulative distribution function (CDF) of the assumed distribution** to find the probability of a bulb lasting more than 648 hours. 

**2. Finding the Probability of Getting 12 Non-Defective Items**

* **Determine the probability of a single item being non-defective:**
    * If the mean of defective items is 0.72, the mean of non-defective items is 1 - 0.72 = 0.28.

* **Use the Poisson distribution:** 
    * The Poisson distribution models the probability of a given number of events occurring within a fixed interval of time or space, given the average rate of occurrence.

    * **Calculate the probability of getting 12 non-defective items using the Poisson probability mass function.**

**3. Testing for Independence in the TV Ownership Data**

* **Perform a Chi-Square test of independence:**

    * **Create a contingency table** summarizing the data.
    * **Calculate the expected frequencies** for each cell in the table under the assumption of independence.
    * **Calculate the Chi-Square statistic:** 
        * Sum the squared differences between the observed and expected frequencies, divided by the expected frequencies.
    * **Determine the degrees of freedom:**
        * (Number of rows - 1) * (Number of columns - 1)
    * **Find the critical Chi-Square value** based on the degrees of freedom and the chosen significance level (5% in this case).
    * **Compare the calculated Chi-Square statistic to the critical Chi-Square value.**
    * **Make a decision:** 
        * If the calculated Chi-Square statistic is greater than the critical Chi-Square value, reject the null hypothesis of independence. 
        * Otherwise, fail to reject the null hypothesis.

**Note:**

* This is a general outline. You'll need to use statistical software (like R, Python, or Excel) or statistical tables to perform the calculations and make the final decisions.
* Ensure you understand the underlying statistical concepts and assumptions before proceeding with the calculations.

I hope this helps! Let me know if you have any further questions or need more specific guidance on any of the parts.