Question 2 (20 marks) A factory produces cylindrical metal bar. The production process can be modeled by normal distribu

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Question 2 (20 marks)
A factory produces cylindrical metal bar. The production process
can be modeled by normal distribution with mean length of 11 cm and
standard deviation of 0.25 cm.
(a) What is the probability that a randomly selected cylindrical
metal bar has a length longer than 10.5 cm?
(b) There is 14% chance that a randomly selected cylindrical
metal bar has a length longer than K. What is the value of K?
(c) The production cost of a metal bar is $80 per cm plus a
basic cost of $100. Find the mean, median, standard deviation,
variance, and 86th percentile of the production cost of a metal
bar.
(d) Write a short paragraph (about 30 – 50 words) to summarize
the production cost of a metal bar. (The summary needs to include
all summary statistics found in part (c)). (e) In order to minimize
the chance of the production cost of a metal bar to be more
expensive than $1000, the senior manager decides to adjust the
production process of the metal bar. The mean length is fixed and
can’t be changed while the standard deviation can be adjusted.
Should the process standard deviation be adjusted to (I) a higher
level than 0.25 cm, or (II) a lower level than 0.25 cm? (Write down
your suggestion, no explanation is needed in part (e)).