A company wishes to set control limits for monitoring the direct labour time to

Question

A company wishes to set control limits for monitoring the direct labour time to produce an important product. Over the past the mean time has been 30 hours with a standard deviation of 10 hours and is believed to be normally distributed. The company proposes to collect random samples of 64 observations to monitor labour time.

If management wishes to establish x-bar control limits covering the 95% confidence interval, calculate the appropriate UCL and LCL.

Calculate the control limits for each of the 3 alternatives.

Which procedure will provide the narrowest control limits? What are they?

If management wishes to use smaller samples of 16 observations; calculate the control limits covering the 95% confidence interval.

Management is considering three alternative procedures in order to maintain tighter control over labour time:

Sampling more frequently using 16 observations and setting confidence intervals of 90%

Maintaining 95% confidence intervals and increasing sample size to 64 observations

Setting 95% confidence intervals and using sample sizes of 36 observations.

Explanation / Answer

We have the mean of sample means (m) that is 30 hours. The standard deviation (sigma) is 10. At 95% confidence level the z value is 1.96. At 90% it is 1.64. (These values are obtained from a standard z-table). Since the sample size is 64 observation we first need to adjust the standard deviation by dividing the sigma by number of sample size. This makes the adjusted standard deviation (SD) = 10/64 = 0.15 for 64 samples. In case of 36 samples the SD is 0.27 and at 16 samples the SD is 0.625.

UCL and LCL are calculated using the formula

UCL = m + z*SD

LCL = m – z*SD

Hence at 95% confidence interval with 64 sample size

UCL = 30 + 1.96*0.15 = 30.294

LCL = 30 – 1.96*0.15= 29.706

Range = 0.588

At 95% confidence interval with 36 sample size

UCL = 30 + 1.96*0.27 = 30.529

LCL = 30 - 1.96*0.27 = 29.470

Range = 1.059

At 90% confidence interval with 16 sample size

UCL = 30 + 1.64*0.625 = 31.025

LCL = 30 – 1.64*0.625 = 28.975

Range = 2.05

64 samples and 95% confidence interval provides the narrowest control limit. The UCL, LCL and the range is mentioned above.

At 95% confidence interval and 16 samples the

UCL = 30 + 1.96*0.625 =31.225

LCL = 30 – 1.96*0.625 = 28.775

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