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 5 kettles, all of which receive dye liquor from a common source. Past data show that pH varies according to a Normal distribution with μ = 4.69 and σ = 0.118. 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 5. 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,

= 4.69

= 0.118

n = 5

Using Empirical rule,

= 4.69 and

= / n = 0.118 / 5 = 0.0528

Using Empirical rule,

P(  - 3 < X <   + 3 ) = 99%

P(4.69 - 3 * 0.0528 < X < 4.69 + 3 * 0.0528) = 99%

P(4.532 < X < 4.848) = 99%

Smaller values = 4.532

Larger values = 4.848