Statistics questions - help!!!?

Question

A BD Company purchases chains from a third party supplier. The chains are required to be within 0.25” of 54 inches in order to work properly. If the chains are too slack or too tight the derailleur won't work. Chains are supplied by the BJ Company and the individual links are not exactly uniform. They fluctuate around a mean length of 0.5”, with a standard deviation of 0.01”.

- How many links should be strung together to form a chain?

- What proportion of the chains would meet BD's standards?

- Supposed BJ Company wished to reduce the proportion of chains that failed to meet Bridgestone’s standards to one third the current value to what value would they have to change the standard deviation of the length of the individual chain links?

Any ideas?!

Thanks in advance!

Answer 1

- How many links should be strung together to form a chain?

Number of links = 54/0.5 = 108
sd = 108 * 0.01 = 1.08

So mean length (μ) of 108 links = 54" with a sd (σ) of 1.08"

- What proportion of the chains would meet BD's standards?

z = (x - μ)/σ + 0.5
= (54.25 - 54)/1.08 + 0.5
= ~0.7315

From a table of z-scores establish the proportion that would lie above the mean and double it for those that lie below the mean.

P(-∞ to 0.7315) ≈ 0.617 (Ref Table 1)
Therefore P (-0.7315 to 0.7315)
≈ (0.617 - 0.5)*2
= 0.117 *2
= 0. 234
ie 23.4% of the chains will meet the standards and 76.6% of chains fail

- Supposed BJ Company wished to reduce the proportion of chains that failed to meet Bridgestone’s standards to one third the current value to what value would they have to change the standard deviation of the length of the individual chain links?

if only 1/3 fail (1/6 too long and 1/6 too short) then 2/3 pass and so want to establish z for p ≈ 0.5 + 0.333 = 0.833

From table (See Ref 1)
z ≈ 0.7975

So 0.7975 = (54.25 - 54)/σ + 0.5
0.25/σ = 0.2975
σ = 0.25/0.2975
= 0.84 inches for the 108 links
= 0.0039 inches per link