Multiple probability calculations

3 In a certain town, the time, \(X\) hours, for which people watch television in a week has a normal distribution with mean 15.8 hours and standard deviation 4.2 hours.

1. Find the probability that a randomly chosen person from this town watches television for less than 21 hours in a week.
2. Find the value of \(k\) such that \(\mathrm { P } ( X < k ) = 0.75\).

2 The lengths of the tails of adult raccoons of a certain species are normally distributed with mean 28 cm and standard deviation 3.3 cm .

1. Find the probability that a randomly chosen adult raccoon of this species has a tail length between 23 cm and 35 cm .
   The masses of adult raccoons of this species are normally distributed with mean 8.5 kg and standard deviation \(\sigma \mathrm { kg } .75 \%\) of adult raccoons of this species have mass greater than 7.6 kg .
2. Find the value of \(\sigma\).

3 The weights of apples of a certain variety are normally distributed with mean 82 grams. \(22 \%\) of these apples have a weight greater than 87 grams.

1. Find the standard deviation of the weights of these apples.
2. Find the probability that the weight of a randomly chosen apple of this variety differs from the mean weight by less than 4 grams.

3 The time spent by shoppers in a large shopping centre has a normal distribution with mean 96 minutes and standard deviation 18 minutes.

1. Find the probability that a shopper chosen at random spends between 85 and 100 minutes in the shopping centre.
   \(88 \%\) of shoppers spend more than \(t\) minutes in the shopping centre.
2. Find the value of \(t\).

1 The times taken to swim 100 metres by members of a large swimming club have a normal distribution with mean 62 seconds and standard deviation 5 seconds.

1. Find the probability that a randomly chosen member of the club takes between 56 and 66 seconds to swim 100 metres.
2. \(13 \%\) of the members of the club take more than \(t\) minutes to swim 100 metres. Find the value of \(t\).

2 The weights of large bags of pasta produced by a company are normally distributed with mean 1.5 kg and standard deviation 0.05 kg .

1. Find the probability that a randomly chosen large bag of pasta weighs between 1.42 kg and 1.52 kg .
   The weights of small bags of pasta produced by the company are normally distributed with mean 0.75 kg and standard deviation \(\sigma \mathrm { kg }\). It is found that \(68 \%\) of these small bags have weight less than 0.9 kg .
2. Find the value of \(\sigma\).

3 In Molimba, the heights, in cm , of adult males are normally distributed with mean 176 cm and standard deviation 4.8 cm .

1. Find the probability that a randomly chosen adult male in Molimba has a height greater than 170 cm .
   60\% of adult males in Molimba have a height between 170 cm and \(k \mathrm {~cm}\), where \(k\) is greater than 170 .
2. Find the value of \(k\), giving your answer correct to 1 decimal place.

3

1. The height of sunflowers follows a normal distribution with mean 112 cm and standard deviation 17.2 cm . Find the probability that the height of a randomly chosen sunflower is greater than 120 cm .
2. When a new fertiliser is used, the height of sunflowers follows a normal distribution with mean 115 cm . Given that \(80 \%\) of the heights are now greater than 103 cm , find the standard deviation.

4
\includegraphics[max width=\textwidth, alt={}, center]{1a10471c-5810-44ca-9353-c2c76e190a2b-2_542_876_1425_632} The random variable \(X\) has a normal distribution with mean 4.5. It is given that \(\mathrm { P } ( X > 5.5 ) = 0.0465\) (see diagram).

1. Find the standard deviation of \(X\).
2. Find the probability that a random observation of \(X\) lies between 3.8 and 4.8.

4 Packets of rice are filled by a machine and have weights which are normally distributed with mean 1.04 kg and standard deviation 0.017 kg .

1. Find the probability that a randomly chosen packet weighs less than 1 kg .
2. How many packets of rice, on average, would the machine fill from 1000 kg of rice? The factory manager wants to produce more packets of rice. He changes the settings on the machine so that the standard deviation is the same but the mean is reduced to \(\mu \mathrm { kg }\). With this mean the probability that a packet weighs less than 1 kg is 0.0388 .
3. Find the value of \(\mu\).
4. How many packets of rice, on average, would the machine now fill from 1000 kg of rice?

7 In Jimpuri the weights, in kilograms, of boys aged 16 years have a normal distribution with mean 61.4 and standard deviation 12.3.

1. Find the probability that a randomly chosen boy aged 16 years in Jimpuri weighs more than 65 kilograms.
2. For boys aged 16 years in Jimpuri, \(25 \%\) have a weight between 65 kilograms and \(k\) kilograms, where \(k\) is greater than 65 . Find \(k\).
   In Brigville the weights, in kilograms, of boys aged 16 years have a normal distribution. \(99 \%\) of the boys weigh less than 97.2 kilograms and \(33 \%\) of the boys weigh less than 55.2 kilograms.
3. Find the mean and standard deviation of the weights of boys aged 16 years in Brigville.

7

1. The time, \(X\) hours, for which students use a games machine in any given day has a normal distribution with mean 3.24 hours and standard deviation 0.96 hours.
   1. On how many days of the year ( 365 days) would you expect a randomly chosen student to use a games machine for less than 4 hours?
   2. Find the value of \(k\) such that \(\mathrm { P } ( X > k ) = 0.2\).
   3. Find the probability that the number of hours for which a randomly chosen student uses a games machine in a day is within 1.5 standard deviations of the mean.
2. The variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\), where \(4 \sigma = 3 \mu\) and \(\mu \neq 0\). Find the probability that a randomly chosen value of \(Y\) is positive.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 The shortest time recorded by an athlete in a 400 m race is called their personal best (PB). The PBs of the athletes in a large athletics club are normally distributed with mean 49.2 seconds and standard deviation 2.8 seconds.

1. Find the probability that a randomly chosen athlete from this club has a PB between 46 and 53 seconds.
2. It is found that \(92 \%\) of athletes from this club have PBs of more than \(t\) seconds. Find the value of \(t\).
   Three athletes from the club are chosen at random.
3. Find the probability that exactly 2 have PBs of less than 46 seconds.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 The heights of students at the Mainland college are normally distributed with mean 148 cm and standard deviation 8 cm .

1. The probability that a Mainland student chosen at random has a height less than \(h \mathrm {~cm}\) is 0.67 . Find the value of \(h\).
   120 Mainland students are chosen at random.
2. Find the number of these students that would be expected to have a height within half a standard deviation of the mean.

3 The number of minutes, \(X\), for which a particular model of laptop computer will run on battery power is Normally distributed with mean 115.3 and standard deviation 21.9.

1. (A) Find \(\mathrm { P } ( X < 120 )\).
   (B) Find \(\mathrm { P } ( 100 < X < 110 )\).
   (C) Find the value of \(k\) for which \(\mathrm { P } ( X > k ) = 0.9\). The number of minutes, \(Y\), for which a different model of laptop computer will run on battery power is known to be Normally distributed with mean \(\mu\) and standard deviation \(\sigma\).
2. Given that \(\mathrm { P } ( Y < 180 ) = 0.7\) and \(\mathrm { P } ( Y < 140 ) = 0.15\), find the values of \(\mu\) and \(\sigma\).
3. Find values of \(a\) and \(b\) for which \(\mathrm { P } ( a < Y < b ) = 0.95\).

3 The random variable \(X\) represents the reaction times, in milliseconds, of men in a driving simulator. \(X\) is Normally distributed with mean 355 and standard deviation 52.

1. Find
   (A) \(\mathrm { P } ( X < 325 )\),
   (B) \(\mathrm { P } ( 300 < X < 400 )\).
2. Find the value of \(k\) for which \(\mathrm { P } ( X < k ) = 0.2\). It is thought that women may have a different mean reaction time from men. In order to test this, a random sample of 25 women is selected. The mean reaction time of these women in the driving simulator is 344 milliseconds. You may assume that women's reaction times are also Normally distributed with standard deviation 52 milliseconds. A hypothesis test is carried out to investigate whether women have a different mean reaction time from men.
3. Carry out the test at the \(5 \%\) significance level.

3 The wing lengths of native English male blackbirds, measured in mm , are Normally distributed with mean 130.5 and variance 11.84.

1. Find the probability that a randomly selected native English male blackbird has a wing length greater than 135 mm .
2. Given that \(1 \%\) of native English male blackbirds have wing length more than \(k \mathrm {~mm}\), find the value of \(k\).
3. Find the probability that a randomly selected native English male blackbird has a wing length which is 131 mm correct to the nearest millimetre. It is suspected that Scandinavian male blackbirds have, on average, longer wings than native English male blackbirds. A random sample of 20 Scandinavian male blackbirds has mean wing length 132.4 mm . You may assume that wing lengths in this population are Normally distributed with variance \(11.84 \mathrm {~mm} ^ { 2 }\).
4. Carry out an appropriate hypothesis test, at the \(5 \%\) significance level.
5. Discuss briefly one advantage and one disadvantage of using a \(10 \%\) significance level rather than a \(5 \%\) significance level in hypothesis testing in general.

3 Many types of computer have cooling fans. The random variable \(X\) represents the lifetime in hours of a particular model of cooling fan. \(X\) is Normally distributed with mean 50600 and standard deviation 3400.

1. Find \(\mathrm { P } ( 50000 < X < 55000 )\).
2. The manufacturers claim that at least \(95 \%\) of these fans last longer than 45000 hours. Is this claim valid?
3. Find the value of \(h\) for which \(99.9 \%\) of these fans last \(h\) hours or more.
4. The random variable \(Y\) represents the lifetime in hours of a different model of cooling fan. \(Y\) is Normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is known that \(\mathrm { P } ( Y < 60000 ) = 0.6\) and \(\mathrm { P } ( Y > 50000 ) = 0.9\). Find the values of \(\mu\) and \(\sigma\).
5. Sketch the distributions of lifetimes for both types of cooling fan on a single diagram.

7. A machine fills bottles with water. The volume of water delivered by the machine to a bottle is \(X \mathrm { ml }\) where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) One of these bottles of water is selected at random. Given that \(\mu = 503\) and \(\sigma = 1.6\)

1. find
   1. \(\mathrm { P } ( X > 505 )\)
   2. \(\mathrm { P } ( 501 < X < 505 )\)
2. Find \(w\) such that \(\mathrm { P } ( 1006 - w < X < w ) = 0.9426\) Following adjustments to the machine, the volume of water delivered by the machine to a bottle is such that \(\mu = 503\) and \(\sigma = q\) Given that \(\mathrm { P } ( X < r ) = 0.01\) and \(\mathrm { P } ( X > r + 6 ) = 0.05\)
3. find the value of \(r\) and the value of \(q\)
   

1. 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.

1. Show that the probability that the weight of this fruit is less than 240 grams is 0.8849
2. Find the probability that the weight of this fruit is between 190 grams and 240 grams.
3. 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.
4. 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,
5. calculate the mean and standard deviation of the weights of this variety of fruit.

1. A machine fills bottles with water. The volume of water delivered by the machine to a bottle is \(X \mathrm { ml }\) where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\)

One of these bottles of water is selected at random.
Given that \(\mu = 503\) and \(\sigma = 1.6\)

1. find
   1. \(\mathrm { P } ( X > 505 )\)
   2. \(\mathrm { P } ( 501 < X < 505 )\)
2. Find \(w\) such that \(\mathrm { P } ( 1006 - w < X < w ) = 0.9426\) Following adjustments to the machine, the volume of water delivered by the machine to a bottle is such that \(\mu = 503\) and \(\sigma = q\) Given that \(\mathrm { P } ( X < r ) = 0.01\) and \(\mathrm { P } ( X > r + 6 ) = 0.05\)
3. find the value of \(r\) and the value of \(q\)
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6. The random variable \(X\) has a normal distribution with mean 30 and standard deviation 5 .

1. Find \(\mathrm { P } ( X < 39 )\).
2. Find the value of \(d\) such that \(\mathrm { P } ( X < d ) = 0.1151\)
3. Find the value of \(e\) such that \(\mathrm { P } ( X > e ) = 0.1151\)
4. Find \(\mathrm { P } ( d < X < e )\).

3. The continuous random variable \(Y\) is normally distributed with mean 100 and variance 256 .

1. Find \(\mathrm { P } ( Y < 80 )\).
2. Find \(k\) such that \(\mathrm { P } ( 100 - k \leq Y \leq 100 + k ) = 0.516\).

1. A scientist found that the time taken, \(M\) minutes, to carry out an experiment can be modelled by a normal random variable with mean 155 minutes and standard deviation 3.5 minutes.

Find

1. \(\mathrm { P } ( M > 160 )\).
2. \(\mathrm { P } ( 150 \leqslant M \leqslant 157 )\).
3. the value of \(m\), to 1 decimal place, such that \(\mathrm { P } ( M \leqslant m ) = 0.30\).