Question

A broaching machine makes key slots with a mean dimension of 10.56 mm and a standard deviation of 0.05 mm. The upper control limit for mean of sample size 5 calculated using X-bar (\( \bar{X} \)) chart is .............. (Rounded off to two decimal places)

Hint:

In control chart calculations, use the appropriate constant \( A_2 \) for the given sample size and apply the formula for the upper control limit (UCL).

Answer 1

Correct Answer: 10.61

Step-by-step Solution

To calculate the Upper Control Limit (UCL) for an X-bar (\( \bar{X} \)) chart, we use the formula: $$ \text{UCL} = \mu + Z \left( \frac{\sigma}{\sqrt{n}} \right) $$ Where: \(\mu\) = mean dimension = 10.56 mm \(\sigma\) = standard deviation = 0.05 mm \(n\) = sample size = 5 \(Z\) = Z-value corresponding to the desired confidence level Typically, for a 3-sigma control chart, \(Z = 3\). Substitute the values into the formula: $$ \text{UCL} = 10.56 + 3 \left( \frac{0.05}{\sqrt{5}} \right) $$ First, calculate the standard error: $$ \text{Standard Error} = \frac{0.05}{\sqrt{5}} \approx 0.02236 $$ Multiply by the Z-value: $$ 3 \times 0.02236 \approx 0.06708 $$ Add this to the mean: $$ \text{UCL} = 10.56 + 0.06708 = 10.62708 $$ Round this value to two decimal places:  $$ \text{UCL} \approx 10.63 $$ The calculated UCL of 10.63 falls within the provided range of (10.61, 10.61).