Find standard deviation from probability

2 The weights of bags of sugar are normally distributed with mean 1.04 kg and standard deviation \(\sigma \mathrm { kg }\). In a random sample of 2000 bags of sugar, 72 weighed more than 1.10 kg . Find the value of \(\sigma\).

1 The height of maize plants in Mpapwa is normally distributed with mean 1.62 m and standard deviation \(\sigma \mathrm { m }\). The probability that a randomly chosen plant has a height greater than 1.8 m is 0.15 . Find the value of \(\sigma\).

5 The lengths of videos of a certain popular song have a normal distribution with mean 3.9 minutes. \(18 \%\) of these videos last for longer than 4.2 minutes.

1. Find the standard deviation of the lengths of these videos.
2. Find the probability that the length of a randomly chosen video differs from the mean by less than half a minute.
   The lengths of videos of another popular song have a normal distribution with the same mean of 3.9 minutes but the standard deviation is twice the standard deviation in part (i). The probability that the length of a randomly chosen video of this song differs from the mean by less than half a minute is denoted by \(p\).
3. Without any further calculation, determine whether \(p\) is more than, equal to, or less than your answer to part (ii). You must explain your reasoning.

1 In a normal distribution with mean 9.3, the probability of a randomly chosen value being greater than 5.6 is 0.85 . Find the standard deviation.

1 Packets of tea are labelled as containing 250 g . The actual weight of tea in a packet has a normal distribution with mean 260 g and standard deviation \(\sigma \mathrm { g }\). Any packet with a weight less than 250 g is classed as 'underweight'. Given that \(1 \%\) of packets of tea are underweight, find the value of \(\sigma\). [3]

2 A continuous random variable has a normal distribution with mean 25.0 and standard deviation \(\sigma\). The probability that any one observation of the random variable is greater than 20,0 is 0.75 . Find the value of \(\sigma\).

4 The random variable \(G\) has mean 20.0 and standard deviation \(\sigma\). It is given that \(\mathrm { P } ( G > 15.0 ) = 0.6\). Assume that \(G\) is normally distributed.

1. (a) Find the value of \(\sigma\).
   (b) Given that \(\mathrm { P } ( G > g ) = 0.4\), find the value of \(\mathrm { P } ( G > 2 g )\).
2. It is known that no values of \(G\) are ever negative. State with a reason what this tells you about the assumption that \(G\) is normally distributed.

1. In a factory, a machine is used to fill bags of rice. The weights of bags of rice are modelled using a normal distribution with mean 250 g .

Only \(1 \%\) of the bags of rice weigh more than 256 g .

1. Write down the percentage of bags of rice with weights between 244 g and 256 g .
2. Find the standard deviation of the weights of the bags of rice. An inspection consists of selecting a bag of rice at random and checking if its weight is within 4 g of the mean. If the weight is more than 4 g away from the mean, then a second bag of rice is selected at random and checked. If the weight of each of the 2 bags of rice is more than 4 g away from the mean, then the machine is shut down.
3. Find the probability that the machine is shut down following an inspection.

2 The mass, in kilograms, of a packet of flour is a normally distributed random variable with mean 1.03 and variance \(\sigma ^ { 2 }\). Given that \(5 \%\) of packets have mass less than 1.00 kg , find the percentage of packets with mass greater than 1.05 kg .

1. A manufacturer uses a machine to make metal rods.

The length of a metal rod, \(L \mathrm {~cm}\), is normally distributed with

• a mean of 8 cm
• a standard deviation of \(x \mathrm {~cm}\)

Given that the proportion of metal rods less than 7.902 cm in length is \(2.5 \%\)

1. show that \(x = 0.05\) to 2 decimal places.
2. Calculate the proportion of metal rods that are between 7.94 cm and 8.09 cm in length. The cost of producing a single metal rod is 20p
   A metal rod
   • where \(L < 7.94\) is sold for scrap for 5 p
   • where \(7.94 \leqslant L \leqslant 8.09\) is sold for 50 p
   • where \(L > 8.09\) is shortened for an extra cost of 10 p and then sold for 50 p
   • Calculate the expected profit per 500 of the metal rods.
   Give your answer to the nearest pound. The same manufacturer makes metal hinges in large batches.
   The hinges each have a probability of 0.015 of having a fault.
   A random sample of 200 hinges is taken from each batch and the batch is accepted if fewer than 6 hinges are faulty. The manufacturer's aim is for 95\% of batches to be accepted.
3. Explain whether the manufacturer is likely to achieve its aim.

3. A drinks machine dispenses coffee into cups. A sign on the machine indicates that each cup contains 50 ml of coffee. The machine actually dispenses a mean amount of 55 ml per cup and \(10 \%\) of the cups contain less than the amount stated on the sign. Assuming that the amount of coffee dispensed into each cup is normally distributed find

1. the standard deviation of the amount of coffee dispensed per cup in ml ,
2. the percentage of cups that contain more than 61 ml . Following complaints, the owners of the machine make adjustments. Only \(2.5 \%\) of cups now contain less than 50 ml . The standard deviation of the amount dispensed is reduced to 3 ml . Assuming that the amount of coffee dispensed is still normally distributed,
3. find the new mean amount of coffee per cup.
   (4)

5. A health club lets members use, on each visit, its facilities for as long as they wish. The club's records suggest that the length of a visit can be modelled by a normal distribution with mean 90 minutes. Only \(20 \%\) of members stay for more than 125 minutes.

1. Find the standard deviation of the normal distribution.
2. Find the probability that a visit lasts less than 25 minutes. The club introduce a closing time of 10:00 pm. Tara arrives at the club at 8:00 pm.
3. Explain whether or not this normal distribution is still a suitable model for the length of her visit.

8. A company has a customer services call centre. The company believes that the time taken to complete a call to the call centre may be modelled by a normal distribution with mean 16 minutes and standard deviation \(\sigma\) minutes. Given that \(10 \%\) of the calls take longer than 22 minutes,

1. show that, to 3 significant figures, the value of \(\sigma\) is 4.68
2. Calculate the percentage of calls that take less than 13 minutes. A supervisor in the call centre claims that the mean call time is less than 16 minutes. He collects data on his own call times.
   • \(20 \%\) of the supervisor's calls take more than 17 minutes to complete.
   • \(10 \%\) of the supervisor's calls take less than 8 minutes to complete.
   Assuming that the time the supervisor takes to complete a call may be modelled by a normal distribution,
3. estimate the mean and the standard deviation of the time taken by the supervisor to complete a call.
4. State, giving a reason, whether or not the calculations in part (c) support the supervisor's claim. \section*{9.} A fast food company has a scratchcard competition. It has ordered scratchcards for the competition and requested that \(45 \%\) of the scratchcards be winning scratchcards. A random sample of 20 of the scratchcards is collected from each of 8 of the fast food company's stores. Assuming that \(45 \%\) of the scratchcards are winning scratchcards, calculate the probability that in at least 2 of the 8 stores, 12 or more of the scratchcards are winning scratchcards.
   [0pt] [Total 5 marks] END OF TEST