- The weights, \(X\) grams, of a particular variety of fruit are normally distributed with
$$X \sim \mathrm {~N} \left( 210,25 ^ { 2 } \right)$$
A fruit of this variety is selected at random.
- Show that the probability that the weight of this fruit is less than 240 grams is 0.8849
- Find the probability that the weight of this fruit is between 190 grams and 240 grams.
- Find the value of \(k\) such that \(\mathrm { P } ( 210 - k < X < 210 + k ) = 0.95\)
A wholesaler buys large numbers of this variety of fruit and classifies the lightest \(15 \%\) as small.
- Find the maximum weight of a fruit that is classified as small.
You must show your working clearly.
The weights, \(Y\) grams, of a second variety of this fruit are normally distributed with
$$Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$$
Given that \(5 \%\) of these fruit weigh less than 152 grams and \(40 \%\) weigh more than 180 grams,
- calculate the mean and standard deviation of the weights of this variety of fruit.