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4) The weights, in grams, of the contents of mackerel filets are normally distributed with mean \( \mu \) and standard deviation 2.5 . The value of \( \mu \) may be adjusted as required. a) Find the proportion of tins with contents weighing between 125.0 grams and 130.0 grams when \( \mu=129.0 \). b) i) State, without proof, the value of \( \mu \) which would maximize the proportion of tins with contents weighing between 125.0 grams and 130.0 grams. Page 1 of 5 In2022-Semester 2 ii) Find the proportion of tins with contents weighing between 125.0 grams and 130.0 grams when \( \mu \) is equal to the value you have specified in part (b)(i). c) Find, to one decimal place, the value of \( \mu \) such that \( 99 \% \) of the tins have contents weighing more than 125.0 grams. d) The normal distribution provides a good model for many continuous distributions which arise in the production process or in nature. Explain why the central Limit Theorem provides another reason for the importance of the normal distribution.

          4) The weights, in grams, of the contents of mackerel filets are normally distributed with mean \( \mu \) and standard deviation 2.5 . The value of \( \mu \) may be adjusted as required.
a) Find the proportion of tins with contents weighing between 125.0 grams and 130.0 grams when \( \mu=129.0 \).
b) i) State, without proof, the value of \( \mu \) which would maximize the proportion of tins with contents weighing between 125.0 grams and 130.0 grams.
Page 1 of 5
In2022-Semester 2
ii) Find the proportion of tins with contents weighing between 125.0 grams and 130.0 grams when \( \mu \) is equal to the value you have specified in part (b)(i).
c) Find, to one decimal place, the value of \( \mu \) such that \( 99 \% \) of the tins have contents weighing more than 125.0 grams.
d) The normal distribution provides a good model for many continuous distributions which arise in the production process or in nature. Explain why the central Limit Theorem provides another reason for the importance of the normal distribution.

4) The weights, in grams, of the contents of mackerel filets are normally distributed with mean μ and standard deviation 2.5 . The value of μ may be adjusted as required.
a) Find the proportion of tins with contents weighing between 125.0 grams and 130.0 grams when μ=129.0.
b) i) State, without proof, the value of μ which would maximize the proportion of tins with contents weighing between 125.0 grams and 130.0 grams.
Page 1 of 5
In2022-Semester 2
ii) Find the proportion of tins with contents weighing between 125.0 grams and 130.0 grams when μ is equal to the value you have specified in part (b)(i).
c) Find, to one decimal place, the value of μ such that 99 % of the tins have contents weighing more than 125.0 grams.
d) The normal distribution provides a good model for many continuous distributions which arise in the production process or in nature. Explain why the central Limit Theorem provides another reason for the importance of the normal distribution.

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