Question

An on-line production of printed-circuit boards (PCBs) involves ten successive sequences (i = 1, 2, 3, …, 10) of component-placement/assembly on the board. In a quality-control effort, the board is tested for its integrity in each of ten successive assembly-tasks. It is presumed that the reliability of final, completed PCB depends on possible, failure-proneness associated with each (successive) assembly-task involved. Table given below shows the number of PCBs (n_i) per successive batch tested. Also shown in the table, is the random number of boards failed in the assembly-line (depicting the set of random variables, RV: x_i) in each of the ten successive batches tested.

	Successively done testing sequence (i = 1 to 10 batches)
	#1	#2	#3	#4	#5	#6	#7	#8	#9	#10
Number of PCBs tested (ni) in each assembly batch	1000	1500	1750	2000	2250	3000	1250	750	2100	1300
Number of failed boards: RV (xi) in each batch tested	10	50	25	33	40	38	20	45	52	44

Determine the following:

1. Most appropriate average performance measure expressed in terms of AM or HM or GM values of the failures involved: Why?
2. Expected mean: E[.], median, mode, variance and standard deviation (s) values of the RV
3. Skewness of the RV expressed in terms of Pearson-1 skewness coefficient; and, specify the type of skewness in terms of relevant parameters that decide the type (left or right-handed) of asymmetry (based on the positions of average measures marked)
4. Plot (discrete) probability density function (pdf) as a function of RV and construct the associated cdf graph. Marking the mode value (x_Mo) on the cdf plot, indicate the range of RV from {x_i = (x_Mo − s)} to {x_i = (x_Mo + s)} with respect to the mode and estimate approximate value of the associated cumulative probability in that range of RV
5. Using the cdf-plot, determine the cumulative probability of RV falling in excess of: x_i ≥ (E[.] + s) and decide on approximate value of this tail-end cumulative probability of the RV

(Answer hints: AM ≈ 35.5; E[.] ≈ 36.9; and, P_sk-1 ≈ − 0.094)

Answer 1

R code and Excel file in which I have done calculation and make plot.

All the calculation are completed with help Excel and graph are made with help of R software.

(i). AM = 36.92, GM = 34.47 and Mode = 38.

We can see the value of AM, GM and Mode are approximately same that’s why these measures are appropriate measures of average.

(ii) Mean(x) = 36.92, Median(x) = 40, Mode(x) = 38, Var(x) = 133.73 and Sd(x) = 11.56 .

(iii) Pearson-1 skewness of coefficient = -0.094 (Negatively skewed or left skewed)

1. If mean is greater than mode then positively skewed.
2. If mean is equal to mode then symmetric.
3. If mean is less than mode then negatively skewed.

(iv). Range = (26.44, 49.56)

The approximate value associated cumulative distribution in this range = 0.5

(v). P(x > 48.48) = 0.2 (approximately)