SOLUTION: Assignment 3 1. Wick Ltd. manufactures washers. One of the key quality measures is the thickness of the washers. A sample of 5 washers in 7 different groups were recorded as shown

Question 1192645: Assignment 3
1. Wick Ltd. manufactures washers. One of the key quality measures is the thickness of the washers. A sample of 5 washers in 7 different groups were recorded as shown in the table.

Thickness (millimetres)

1
2
3
4
5
1
2.8
3.3
3.0
3.1
2.9
2
3.5
3.4
3.2
2.9
3.3
3
3.1
3.4
3.3
3.1
3.4
4
3.1
3.1
3.1
2.9
3.2
5
3.2
3.2
2.8
2.7
3.5
6
3.3
3.4
3.1
2.9
3.4
7
3.1
3.5
3.2
3.0
3.6

Use the data to construct the chart and R-chart

2. Two machines P and Q are used to produce bags of cement of masses in kilogrammes shown in the table.

Machine P
50
51
48
50
51
52
53
51
51
Machine Q
54
49
56
47
50
51
52
53

Test if there is a difference between the two machines.
C.A.T 2
1. The probability that a life bulb will have a life time of more than 682 hours is 0.9788. The probability that a bulb will have a life time of more than 703 hours is 0.0051. Find the probability that a bulb will last for more than 648 hours.
2. The mean and variance of defective items is 0.72 and 0.6876. Find the probability of getting 12 non-defective items.

3. A random sample of householders were classified using the characteristics namely Digital TV and Smart TV and results recorded as follows:

Owner occupier
Council Tenant
Private Tenant
Smart TV
94
67
53
Digital TV
130
111
45
Black/ White TV
80
65
26

Test for independent at 5%.
Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
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.

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