Question 1 (10 marks) A salad shop is selling fruit cups. Each fruit cup consists of two types of fruit, strawberries an

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Question 1 (10 marks)
A salad shop is selling fruit cups. Each fruit cup consists of two
types of fruit, strawberries and blue berries. The weight of
strawberries in a fruit cup is normally distributed with mean 160
grams and standard deviation 10 grams. The weight of blue berries
in a fruit cup is normally distributed with mean μ grams
and standard deviation σ grams. The weight of
strawberries and blue berries are independent, and it is known that
the weight of a fruit cup with average of 300 grams and standard
deviation of 15 grams.
(a) Find the values of μ and σ.
(b) The weights of the middle 96.6% of fruit cups are
between (300 – K, 300 + K) grams. Find the value of
K.
(c) The weights of the middle 96.6% of fruit cups are
between (L1, L2) grams. Find the values of L1 and L2.
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)).