Light bulbs are tested for their luminance, with the intensity of brightness desired to be within a certain range. Random samples of five bulbs are chosen from the output and their luminance values measured. The sample mean X^‾ and standard deviation s are found. After 30 samples, the following summary information is obtained:

Σ³⁰_i=1 X^‾_i ^{= 2550,}

Σ³⁰_i=1 S_i = 195

The specifications are 90±15 lumens.

(a) Find the control limits for the X^‾- and s-charts.

(b) Assuming that the process in control, estimate the process mean and process standard deviation.

(c) Find the process capability indices C_p and C_pk and comment on their values.

(d) If the target value is 90 lumens, find the capability indices C_pm and C_pmk.

(e) What proportion of the output is nonconforming, assuming a normal distribution of the quality characteristic?

(f) If the process mean is moved to 88 lumens, what proportion of the output is nonconforming? What are your proposals to improve process performance?