The number of flaws per square yard in a type of carpet material varies with mea

Question

The number of flaws per square yard in a type of carpet material varies with mean 1.3 flaws per square yard and standard deviation 0.8 flaws per square yard. This population distribution cannot be normal, because a count takes only whole-number values. An inspector studies 180 square yards of the material, records the number of flaws found in each square yard, and calculates x, the mean number of flaws per square yard inspected. Use the central limit theorem to find the approximate probability that the mean number of flaws exceeds 1.4 per square yard. (Round your answer to four decimal places.)

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 = 4.66 and = 0.143. 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 689599.7 rule for normal distributions, namely

x ± 3x.

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

Explanation / Answer

1) here std error of mean =std deviation/(n)1/2 =0.8/(180)1/2 =0.0596

therefore probability that the mean number of flaws exceeds 1.4 per square yard =P(X>1.4)=1-P(X<1.4)

=1-P(Z<(1.4-1.3)/0.0596)=1-P(Z<1.6771)=1-0.9532 =0.0468

2) here  std error of mean =std deviation/(n)1/2 =  x =0.143/(5)1/2 =0.0640

smaller value =x - 3x =4.468

(larger value =x + 3x =4.852

please revert for any clarification required

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