Question

The unique colors of the cashmere sweaters your firm makes result from heating undyed yarn in a kettle with a dye liquor. The pH (acidity) of the liquor is critical for regulating dye uptake and hence the final color. There are 7 kettles, all of which receive dye liquor from a common source. Past data show that pH varies according to a Normal distribution with μ = 4.71 and σ = 0.129. You use statistical process control to check the stability of the process. Twice each day, the pH of the liquor in each kettle is measured, giving a sample of size 7. The mean pH x is compared with "control limits" given by the 99.7 part of the 68−95−99.7 rule for normal distributions, namely μx ± 3σx. What are the numerical values of these control limits for x? (Round your answers to three decimal places.)

Smaller value=

Larger value=

Answer 1

Given:

Mean, $\mu_{x} = 4.71$

Standard deviation, $\sigma_{x} = 0.129$

For "control limits" of 99.7, by the 68-95-99.7 rule for normal distributions.

Smaller value of control limit = $\mu_{x} - 3\sigma_{x}$ = 4.71 - 3*0.129 = 4.323

Larger value of control limit = $\mu_{x} + 3\sigma_{x}$ = 4.71 + 3*0.129 = 5.097