3. The weights of Braeburn apples on display in a supermarket, measured in grams, are Normally distributed with mean 210.5 and standard deviation 15.2.
i. Find the probability that a randomly selected apple weighs at least 220 grams.
ii. 80 apples are selected at random.
a) Find the probability that more than 18 of these apples weigh at least 220 grams.
b) Find the expectation and standard deviation for the number of apples that weigh at least 220 grams.
c) State a suitable approximating distribution, including any parameters, for the number of apples that weigh at least 220 grams.
d) Explain why this approximating distribution is suitable. The supermarket also sells Cox's Orange Pippin apples. The weights of these apples, measured in grams, are Normally distributed with mean 185 and standard deviation \(\sigma\).
iii. Given that \(10 \%\) of randomly selected Cox's Orange Pippin apples weigh less than 170 grams, calculate the value of \(\sigma\).