Probability calculation plus find unknown boundary

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 \(t\).

4 In a large population, the systolic blood pressure (SBP) of adults is normally distributed with mean 125.4 and standard deviation 18.6.

1. Find the probability that the SBP of a randomly chosen adult is less than 132.
   The SBP of 12-year-old children in the same population is normally distributed with mean 117. Of these children 88\% have SBP more than 108.
2. Find the standard deviation of this distribution.
   Three adults are chosen at random from this population.
3. Find the probability that each of these three adults has SBP within 1.5 standard deviations of the mean.

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.

6 Tyre pressures on a certain type of car independently follow a normal distribution with mean 1.9 bars and standard deviation 0.15 bars.

1. Find the probability that all four tyres on a car of this type have pressures between 1.82 bars and 1.92 bars.
2. Safety regulations state that the pressures must be between \(1.9 - b\) bars and \(1.9 + b\) bars. It is known that \(80 \%\) of tyres are within these safety limits. Find the safety limits.

7 The times taken to play Beethoven's Sixth Symphony can be assumed to have a normal distribution with mean 41.1 minutes and standard deviation 3.4 minutes. Three occasions on which this symphony is played are chosen at random.

1. Find the probability that the symphony takes longer than 42 minutes to play on exactly 1 of these occasions. The times taken to play Beethoven's Fifth Symphony can also be assumed to have a normal distribution. The probability that the time is less than 26.5 minutes is 0.1 , and the probability that the time is more than 34.6 minutes is 0.05 .
2. Find the mean and standard deviation of the times to play this symphony.
3. Assuming that the times to play the two symphonies are independent of each other, find the probability that, when both symphonies are played, both of the times are less than 34.6 minutes.

7 The length of time a person undergoing a routine operation stays in hospital can be modelled by a normal distribution with mean 7.8 days and standard deviation 2.8 days.

1. Calculate the proportion of people who spend between 7.8 days and 11.0 days in hospital.
2. Calculate the probability that, of 3 people selected at random, exactly 2 spend longer than 11.0 days in hospital.
3. A health worker plotted a box-and-whisker plot of the times that 100 patients, chosen randomly, stayed in hospital. The result is shown below. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{26776153-9477-4155-b5e4-f35e6d33a5ff-3_447_917_767_657} \captionsetup{labelformat=empty} \caption{Days}
   \end{figure} State with a reason whether or not this agrees with the model used in parts (i) and (ii).

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?

4 The time taken to cook an egg by people living in a certain town has a normal distribution with mean 4.2 minutes and standard deviation 0.6 minutes.

1. Find the probability that a person chosen at random takes between 3.5 and 4.5 minutes to cook an egg. \(12 \%\) of people take more than \(t\) minutes to cook an egg.
2. Find the value of \(t\).
3. A random sample of \(n\) people is taken. Find the smallest possible value of \(n\) if the probability that none of these people takes more than \(t\) minutes to cook an egg is less than 0.003 .

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.

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 amount of data, \(X\) megabytes, arriving at an internet server per second during the afternoon is modelled by the Normal distribution with mean 435 and standard deviation 30.

1. Find
   (A) \(\mathrm { P } ( X < 450 )\),
   (B) \(\mathrm { P } ( 400 < X < 450 )\).
2. Find the probability that, during 5 randomly selected seconds, the amounts of data arriving are all between 400 and 450 megabytes. The amount of data, \(Y\) megabytes, arriving at the server during the evening is modelled by the Normal distribution with mean \(\mu\) and standard deviation \(\sigma\).
3. Given that \(\mathrm { P } ( Y < 350 ) = 0.2\) and \(\mathrm { P } ( Y > 390 ) = 0.1\), find the values of \(\mu\) and \(\sigma\).
4. Find values of \(a\) and \(b\) for which \(\mathrm { P } ( a < Y < b ) = 0.95\).

3 In a men's cycling time trial, the times are modelled by the random variable \(X\) minutes which is Normally distributed with mean 63 and standard deviation 5.2.

1. Find $$\begin{aligned} & \text { (A) } \mathrm { P } ( X < 65 ) \text {, } \\ & \text { (B) } \mathrm { P } ( 60 < X < 65 ) \text {. } \end{aligned}$$
2. Find the probability that 5 riders selected at random all record times between 60 and 65 minutes.
3. A competitor aims to be in the fastest \(5 \%\) of entrants (i.e. those with the lowest times). Find the maximum time that he can take. It is suggested that holding the time trial on a new course may result in lower times. To investigate this, a random sample of 15 competitors is selected. These 15 competitors do the time trial on the new course. The mean time taken by these riders is 61.7 minutes. You may assume that times are Normally distributed and the standard deviation is still 5.2 minutes. A hypothesis test is carried out to investigate whether times on the new course are lower.
4. Write down suitable null and alternative hypotheses for the test. Carry out the test at the 5\% significance level.

3 At a vineyard, the process used to fill bottles with wine is subject to variation. The contents of bottles are independently Normally distributed with mean \(\mu = 751.4 \mathrm { ml }\) and standard deviation \(\sigma = 2.5 \mathrm { ml }\).

1. Find the probability that a randomly selected bottle contains at least 750 ml .
2. A case of wine consists of 6 bottles. Find the probability that all 6 bottles in a case contain at least 750 ml .
3. Find the probability that, in a random sample of 25 cases, there are at least 2 cases in which all 6 bottles contain at least 750 ml . It is decided to increase the proportion of bottles which contain at least 750 ml to \(98 \%\).
4. This can be done by changing the value of \(\mu\), but retaining the original value of \(\sigma\). Find the required value of \(\mu\).
5. An alternative is to change the value of \(\sigma\), but retain the original value of \(\mu\). Find the required value of \(\sigma\).
6. Comment briefly on which method might be easier to implement and which might be preferable to the vineyard owners.

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 \(\mathrm { P } ( X > r + 6 ) = 0.05\)
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