Take Home Quiz 5 Name:Beth Bugel Due Hondoy Instructions: You may only work with
ID: 3055258 • Letter: T
Question
Take Home Quiz 5 Name:Beth Bugel Due Hondoy Instructions: You may only work with other current MTH 1151 students or your MTH-115 4-4-18 The mass of chicken eggs produced for market is normally distributed with a mean of 56 gra instructor. NO assistance from other instructors, tutors or others is permitted 1. and a standard deviation of 4 grams. Use the empirical rule to answer (a) and (b). Use your calculator to answer (d) and (e). Show supporting work. (a) About what percent of the eggs would be between 52 g and 60 g? or-Y 6o ms9rams yrams (b) About what percent of the eggs would be at least 48 g?Explanation / Answer
#1.
The empirical rule, also known as the three-sigma rule or the 68-95-99.7 rule, provides a quick estimate of the spread of data in a normal distribution given the mean and standard deviation. Specifically, the empirical rule states that for a normal distribution:
68% of the data will fall within one standard deviation of the mean.
95% of the data will fall within two standard deviations of the mean.
Almost all (99.7%) of the data will fall within three standard deviations of the mean.
a)
52 and 60 lie at 1-sigma level from mean.
Hence 68% of the eggs would be between 52g and 60g
b)
48 lies to the left of the mean at 2-sigma distance.
95% data lies within 2-sigma, however here we only need to exclude the proportion to the left of 2-sigma.
Hence, P(X >= 48) = 1 - 0.025 = 0.975
i.e. 97.5%
c)
Using central limit theorem,
x = mean + z*sigma
x = 56 + 1.5*4
x = 62
Ans: 62g
d)
P(X < 50)
= P(z < (50 - 56)/4)
= P(z < -1.5)
= 0.0668
Hence 6.68% of eggs are less than 50g
e)
P(60 < X < 70)
= P((60 - 56)/4 < z < (70 - 56)/4))
= P(1 < z < 3.5)
= P(z < 3.5) - P(z < 1)
= 0.9998 - 0.8413
= 0.1584
p = 0.1584
required number of eggs = 0.1584*500 = 79.2 dozens
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