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 = 4.91 and σ = 0.131. 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

μ_x ± 3σ_x.

What are the numerical values of these control limits for x? (Round your answers to three decimal places.)

(smaller value)

(larger value)

Answer 1

Solution :

Given that,

= 4.91

= 0.131

n = 7

= 4.91

= / n = 0.131 / 7 = 0.019

Using Empirical rule,

P( - 3< X < + 3) = 99.7%

P(4.91 - 3 * 0.019 < X < 4.91 + 3 * 0.019) = 99.7%

P(4.853 < X < 4.967) = 99.7%

Smallest value = 4.853

Larger value = 4.967