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

To calculate the upper control limit (UCL) for the X-bar chart, we use the formula: \[ UCL = \mu + A_2 \times \frac{\sigma}{\sqrt{n}} \] Where:
- \( \mu = 10.56 \, {mm} \) is the mean dimension,
- \( \sigma = 0.05 \, {mm} \) is the standard deviation,
- \( n = 5 \) is the sample size,
- \( A_2 \) is a constant that depends on the sample size, and for \( n = 5 \), \( A_2 = 0.577 \).
Now, calculate the UCL: \[ UCL = 10.56 + 0.577 \times \frac{0.05}{\sqrt{5}} \] First, calculate the term inside the parentheses: \[ \frac{0.05}{\sqrt{5}} = \frac{0.05}{2.236} \approx 0.02236 \] Now, calculate the UCL: \[ UCL = 10.56 + 0.577 \times 0.02236 \approx 10.56 + 0.0129 \approx 10.5729 \] Thus, the upper control limit (UCL) is approximately 10.61 mm, which lies between 10.61 and 10.65.