Mixed calculations with boundaries

7. Farmer Adam grows potatoes. The weights of potatoes, in grams, grown by Adam are normally distributed with a mean of 140 g and a standard deviation of 40 g . Adam cannot sell potatoes with a weight of less than 92 g .

1. Find the percentage of potatoes that Adam grows but cannot sell. The upper quartile of the weight of potatoes sold by Adam is \(q _ { 3 }\)
2. Find the probability that the weight of a randomly selected potato grown by Adam is more than \(q _ { 3 }\)
3. Find the lower quartile, \(q _ { 1 }\), of the weight of potatoes sold by Adam. Betty selects a random sample of 3 potatoes sold by Adam.
4. Find the probability that one weighs less than \(q _ { 1 }\), one weighs more than \(q _ { 3 }\) and one has a weight between \(q _ { 1 }\) and \(q _ { 3 }\)
   

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5 When a particular make of tennis ball is dropped from a vertical distance of 250 cm on to concrete, the height, \(X\) centimetres, to which it first bounces may be assumed to be normally distributed with a mean of 140 and a standard deviation of 2.5.

1. Determine:
   1. \(\mathrm { P } ( X < 145 )\);
   2. \(\mathrm { P } ( 138 < X < 142 )\).
2. Determine, to one decimal place, the maximum height exceeded by \(85 \%\) of first bounces.
3. Determine the probability that, for a random sample of 4 first bounces, the mean height is greater than 139 cm .

3 The weight, \(X\) grams, of talcum powder in a tin may be modelled by a normal distribution with mean 253 and standard deviation \(\sigma\).

1. Given that \(\sigma = 5\), determine:
   1. \(\mathrm { P } ( X < 250 )\);
   2. \(\mathrm { P } ( 245 < X < 250 )\);
   3. \(\mathrm { P } ( X = 245 )\).
2. Assuming that the value of the mean remains unchanged, determine the value of \(\sigma\) necessary to ensure that \(98 \%\) of tins contain more than 245 grams of talcum powder.
   (4 marks) \includegraphics[max width=\textwidth, alt={}, center]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-07_38_118_440_159}
   \includegraphics[max width=\textwidth, alt={}, center]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-07_40_118_529_159}

2 A garden centre sells bamboo canes of nominal length 1.8 metres. The length, \(X\) metres, of the canes can be modelled by a normal distribution with mean 1.86 and standard deviation \(\sigma\).

1. Assuming that \(\sigma = 0.04\), determine:
   1. \(\mathrm { P } ( X < 1.90 )\);
   2. \(\mathrm { P } ( X > 1.80 )\);
   3. \(\mathrm { P } ( 1.80 < X < 1.90 )\);
   4. \(\mathrm { P } ( X \neq 1.86 )\).
2. It is subsequently found that \(\mathrm { P } ( X > 1.80 ) = 0.98\). Determine the value of \(\sigma\).
   [0pt] [3 marks]
   \includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-06_1529_1717_1178_150}

5 Still mineral water is supplied in 1.5-litre bottles. The actual volume, \(X\) millilitres, in a bottle may be modelled by a normal distribution with mean \(\mu = 1525\) and standard deviation \(\sigma = 9.6\).

1. Determine the probability that the volume of water in a randomly selected bottle is:
   1. less than 1540 ml ;
   2. more than 1535 ml ;
   3. between 1515 ml and 1540 ml ;
   4. not 1500 ml .
2. The supplier requires that only 10 per cent of bottles should contain more than 1535 ml of water. Assuming that there has been no change in the value of \(\sigma\), calculate the reduction in the value of \(\mu\) in order to satisfy this requirement. Give your answer to one decimal place.
3. Sparkling spring water is supplied in packs of six 0.5 -litre bottles. The actual volume in a bottle may be modelled by a normal distribution with mean 508.5 ml and standard deviation 3.5 ml . Stating a necessary assumption, determine the probability that:
   1. the volume of water in each of the 6 bottles from a randomly selected pack is more than 505 ml ;
   2. the mean volume of water in the 6 bottles from a randomly selected pack is more than 505 ml .
      [0pt] [7 marks]

3. The time it takes girls aged 15 to complete an obstacle course is found to be normally distributed with a mean of 21.5 minutes and a standard deviation of 2.2 minutes.

1. Find the probability that a randomly chosen 15 year-old girl completes the course in less than 25 minutes. A 13 year-old girl completes the course in exactly 19 minutes.
2. What percentage of 15 year-old girls would she beat over the course? Anyone completing the course in less than 20 minutes is presented with a certificate of achievement. Three friends all complete the course one afternoon.
3. What is the probability that exactly two of them get certificates?

2 The weight, \(X\) grams, of a particular variety of orange is normally distributed with mean 205 and standard deviation 25.

1. Determine the probability that the weight of an orange is:
   1. less than 250 grams;
   2. between 200 grams and 250 grams.
2. A wholesaler decides to grade such oranges by weight. He decides that the smallest 30 per cent should be graded as small, the largest 20 per cent graded as large, and the remainder graded as medium. Determine, to one decimal place, the maximum weight of an orange graded as:
   1. small;
   2. medium.
3. The weight, \(Y\) grams, of a second variety of orange is normally distributed with mean 175. Given that 90 per cent of these oranges weigh less than 200 grams, calculate the standard deviation of their weights.
   (4 marks)