The lengths of nails manufactured at a factory are normally distributed with a mean length of 4.13 cm and a standard dev

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The lengths of nails manufactured at a factory are normally
distributed with a mean length of 4.13 cm and a standard deviation
of 1.09 cm. For quality control purposes, a random sample of 11
nails is chosen once per day, and measured. Let X be the
random variable representing the mean length of the nails.

a. Fill in the blank, rounding your answers to 2 decimal places if
needed. According to the Central Limit Theorem, X is
approximately normal with a mean of ________ cm and a standard
error of the mean ________ cm

b. Find the z-score associated to a sample with a mean of 4.3 cm,
using the sampling distribution. Round your answer to two decimal
places. ________

c. Find the probability that a randomly selected sample of 11 nails
has a mean length higher than 4.3 cm. Round your answer to 4
decimal places. _________

d. c. Find the probability that a randomly
selected individual nail has a length
higher than 4.3 cm. Round your answer to 4 decimal places.
_________

e. The machine that cuts the nails will be pulled for inspection
and repairs if the sample mean of the quality control sample is in
the bottom 3% or top 3% of possible sample means. What sample
average lengths will trigger a repair? Round your answers to 2
decimal places. _________

The machine will be pulled for repairs if the quality control
sample has a mean length below ________ cm or above
________ cm.