Fixed container with random contents

5 Large packets of rice are packed in cartons, each containing 20 randomly chosen packets. The masses of these packets are normally distributed with mean 1010 g and standard deviation 3.4 g . The masses of the cartons, when empty, are independently normally distributed with mean 50 g and standard deviation 2.0 g .

1. Find the variance of the masses of full cartons.
   Small packets of rice are packed in boxes. The total masses of full boxes are normally distributed with mean 6730 g and standard deviation 15.0 g . The masses of the boxes and cartons are distributed independently of each other.
2. Find the probability that the mass of a randomly chosen full carton is more than three times the mass of a randomly chosen full box.

5 Each box of Fruity Flakes contains \(X\) grams of flakes and \(Y\) grams of fruit, where \(X\) and \(Y\) are independent random variables, having distributions \(\mathrm { N } ( 400,50 )\) and \(\mathrm { N } ( 100,20 )\) respectively. The weight of each box, when empty, is exactly 20 grams. A full box of Fruity Flakes is chosen at random.

1. Find the probability that the total weight of the box and its contents is less than 530 grams.
2. Find the probability that the weight of flakes in the box is more than 4.1 times the weight of fruit in the box.

5 Large packets of sugar are packed in cartons, each containing 12 packets. The weights of these packets are normally distributed with mean 505 g and standard deviation 3.2 g . The weights of the cartons, when empty, are independently normally distributed with mean 150 g and standard deviation 7 g .

1. Find the probability that the total weight of a full carton is less than 6200 g .
   Small packets of sugar are packed in boxes. The total weight of a full box has a normal distribution with mean 3130 g and standard deviation 12.1 g .
2. Find the probability that the weight of a randomly chosen full carton is less than double the weight of a randomly chosen full box.

3 Weights of cups have a normal distribution with mean 91 g and standard deviation 3.2 g . Weights of saucers have an independent normal distribution with mean 72 g and standard deviation 2.6 g . Cups and saucers are chosen at random to be packed in boxes, with 6 cups and 6 saucers in each box. Given that each empty box weighs 550 g , find the probability that the total weight of a box containing 6 cups and 6 saucers exceeds 1550 g .

4 A factory supplies boxes of children's bricks. Each box contains 10 randomly chosen large bricks and 20 randomly chosen small bricks. The masses, in grams, of large and small bricks have the distributions \(\mathrm { N } ( 60,1.2 )\) and \(\mathrm { N } ( 30,0.7 )\) respectively. The mass of an empty box is 8 g . Find the probability that the total weight of a box and its contents is less than 1200 g .

7 Bottles of wine are stacked in racks of 12. The weights of these bottles are normally distributed with mean 1.3 kg and standard deviation 0.06 kg . The weights of the empty racks are normally distributed with mean 2 kg and standard deviation 0.3 kg .

1. Find the probability that the total weight of a full rack of 12 bottles of wine is between 17 kg and 18 kg .
2. Two bottles of wine are chosen at random. Find the probability that they differ in weight by more than 0.05 kg .

4 The weights of a particular variety (A) of tomato are known to be Normally distributed with mean 80 grams and standard deviation 11 grams.

1. Find the probability that a randomly chosen tomato of variety A weighs less than 90 grams. The weights of another variety (B) of tomato are known to be Normally distributed with mean 70 grams. These tomatoes are packed in sixes using packaging that weighs 15 grams.
2. The probability that a randomly chosen pack of 6 tomatoes of variety B , including packaging, weighs less than 450 grams is 0.8463 . Show that the standard deviation of the weight of single tomatoes of variety B is 6 grams, to the nearest gram.
3. Tomatoes of variety A are packed in fives using packaging that weighs 25 grams. Find the probability that the total weight of a randomly chosen pack of variety A is greater than the total weight of a randomly chosen pack of variety B .
4. A new variety (C) of tomato is introduced. The weights, \(c\) grams, of a random sample of 60 of these tomatoes are measured giving the following results. $$\Sigma c = 3126.0 \quad \Sigma c ^ { 2 } = 164223.96$$ Find a \(95 \%\) confidence interval for the true mean weight of these tomatoes.