2.04f Find normal probabilities: Z transformation

5. Rosie keeps bees. The amount of honey, in kg, produced by a hive of Rosie's bees in a season, is modelled by a normal distribution with a mean of 22 kg and a standard deviation of 10 kg .

1. Find the probability that a hive of Rosie's bees produces less than 18 kg of honey in a season. The local bee keepers' club awards a certificate to every hive that produces more than 39 kg of honey in a season, and a medal to every hive that produces more than 50 kg in a season. Given that one of Rosie's bee hives is awarded a certificate
2. find the probability that this hive is also awarded a medal.
   (5) Sam also keeps bees. The amount of honey, in kg, produced by a hive of Sam's bees in a season, is modelled by a normal distribution with mean \(\mu \mathrm { kg }\) and standard deviation \(\sigma \mathrm { kg }\). The probability that a hive of Sam's bees produces less than 28 kg of honey in a season is 0.8413 Only 20\% of Sam's bee hives produce less than 18 kg of honey in a season.
3. Find the value of \(\mu\) and the value of \(\sigma\). Give your answers to 2 decimal places.
   (6)
   
   \includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-11_2261_47_313_37}

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.

7. The weights, \(G\), of a particular breed of gorilla are normally distributed with mean 180 kg and standard deviation 15 kg .

1. Find the proportion of these gorillas whose weights exceed 174 kg .
2. Find, to 1 decimal place, the value of \(k\) such that \(\mathrm { P } ( k < G < 174 ) = 0.3196\) The weights, \(B\), of a particular breed of buffalo are normally distributed with mean 216 kg and standard deviation 30 kg . Given that \(\mathrm { P } ( G > w ) = \mathrm { P } ( B < w ) = p\)
3. 1. find the value of \(w\)
   2. find the value of \(p\) and standard deviation 15 kg .
      1. Find the proportion of these gorillas whose weights exceed 174 kg .
      2. Find, to 1 decimal place, the value of \(k\) such that \(\mathrm { P } ( k < G < 174 ) = 0.3196\)

         Leave blank	Q7

The weights of women boxers in a tournament are normally distributed with mean 64 kg and standard deviation 8 kg .

1. Find the probability that a randomly chosen woman boxer in the tournament weighs less than 51 kg . In the tournament, women boxers who weigh less than 51 kg are classified as lightweight. Ren weighs 49 kg and she has a match against another randomly selected, lightweight woman boxer.
2. Find the probability that Ren weighs less than the other boxer. In the tournament, women boxers who weigh more than \(H \mathrm {~kg}\) are classified as heavyweight. Given that \(10 \%\) of the women boxers in the tournament are classified as heavyweight,
3. find the value of \(H\).
   

3. The weights of packages that arrive at a factory are normally distributed with a mean of 18 kg and a standard deviation of 5.4 kg

1. Find the probability that a randomly selected package weighs less than 10 kg The heaviest 15\% of packages are moved around the factory by Jemima using a forklift truck.
2. Find the weight, in kg , of the lightest of these packages that Jemima will move. One of the packages not moved by Jemima is selected at random.
3. Find the probability that it weighs more than 18 kg A delivery of 4 packages is made to the factory. The weights of the packages are independent.
4. Find the probability that exactly 2 of them will be moved by Jemima.

The lengths, \(L \mathrm {~mm}\), of housefly wings are normally distributed with \(L \sim \mathrm {~N} \left( 4.5,0.4 ^ { 2 } \right)\)

1. Find the probability that a randomly selected housefly has a wing length of less than 3.86 mm .
2. Find
   1. the upper quartile ( \(Q _ { 3 }\) ) of \(L\)
   2. the lower quartile ( \(Q _ { 1 }\) ) of \(L\) A value that is greater than \(Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) or smaller than \(Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) is defined as an outlier.
3. Find these two outlier limits. A housefly is selected at random.
4. Using standardisation, show that the probability that this housefly is not an outlier is 0.993 to 3 decimal places. Given that this housefly is not an outlier,
5. showing your working, find the probability that the wing length of this housefly is greater than 5 mm .

The distance an athlete can throw a discus is normally distributed with mean 40 m and standard deviation 4 m

1. Using standardisation, show that the probability that this athlete throws the discus less than 38.8 m is 0.3821 This athlete enters a discus competition.
   To qualify for the final, they have 3 attempts to throw the discus a distance of more than 38.8 m
   Once they qualify, they do not use any of their remaining attempts.
   Given that they qualified for the final and that throws are independent,
2. find the probability that this athlete qualified for the final on their second throw with a distance of more than 44 m

1. The random variable \(X\) is normally distributed with mean \(\mu\) and variance 36

Given that $$\mathrm { P } ( \mu - 2 k < X < \mu + 2 k ) = 0.6$$

1. find the value of \(k\) The random variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\) Given that $$2 \mu = 3 \sigma ^ { 2 } \quad \text { and } \quad \mathrm { P } \left( \mathrm { Y } > \frac { 3 } { 2 } \mu \right) = 0.0668$$
2. find the value of \(\mu\) and the value of \(\sigma\)

4. The random variable \(Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) Given that \(\mathrm { P } ( Y < 17 ) = 0.6\) find

1. \(\mathrm { P } ( Y > 17 )\)
2. \(\mathrm { P } ( \mu < Y < 17 )\)
3. \(\mathrm { P } ( Y < \mu \mid Y < 17 )\)

7. One event at Pentor sports day is throwing a tennis ball. The distance a child throws a tennis ball is modelled by a normal distribution with mean 32 m and standard deviation 12 m . Any child who throws the tennis ball more than 50 m is awarded a gold certificate.

1. Show that, to 3 significant figures, 6.68\% of children are awarded a gold certificate. A silver certificate is awarded to any child who throws the tennis ball more than \(d\) metres but less than 50 m . Given that 19.1\% of the children are awarded a silver certificate,
2. find the value of \(d\). Three children are selected at random from those who take part in the throwing a tennis ball event.
3. Find the probability that 1 is awarded a gold certificate and 2 are awarded silver certificates. Give your answer to 2 significant figures.

Police measure the speed of cars passing a particular point on a motorway. The random variable \(X\) is the speed of a car. \(X\) is modelled by a normal distribution with mean 55 mph (miles per hour).

1. Draw a sketch to illustrate the distribution of \(X\). Label the mean on your sketch. The speed limit on the motorway is 70 mph . Car drivers can choose to travel faster than the speed limit but risk being caught by the police. The distribution of \(X\) has a standard deviation of 20 mph .
2. Find the percentage of cars that are travelling faster than the speed limit. The fastest \(1 \%\) of car drivers will be banned from driving.
3. Show that the lowest speed, correct to 3 significant figures, for a car driver to be banned is 102 mph . Show your working clearly. Car drivers will just be given a caution if they are travelling at a speed \(m\) such that $$\mathrm { P } ( 70 < X < m ) = 0.1315$$
4. Find the value of \(m\). Show your working clearly.

2. The random variable \(X\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\).

1. Write down 3 properties of the distribution of \(X\). Given that \(\mu = 27\) and \(\sigma = 10\)
2. find \(\mathrm { P } ( 26 < X < 28 )\).

2 A random variable \(C\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). A random sample of 10 observations of \(C\) is obtained, and the results are summarised as $$n = 10 , \Sigma c = 380 , \Sigma c ^ { 2 } = 14602 .$$

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
2. Hence calculate an estimate of the probability that \(C > 40\).

7 The continuous random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The mean of a random sample of \(n\) observations of \(X\) is denoted by \(\bar { X }\). It is given that \(\mathrm { P } ( \bar { X } < 35.0 ) = 0.9772\) and \(\mathrm { P } ( \bar { X } < 20.0 ) = 0.1587\).

1. Obtain a formula for \(\sigma\) in terms of \(n\). Two students are discussing this question. Aidan says "If you were told another probability, for instance \(\mathrm { P } ( \bar { X } > 32 ) = 0.1\), you could work out the value of \(\sigma\)." Binya says, "No, the value of \(\mathrm { P } ( \bar { X } > 32 )\) is fixed by the information you know already."
2. State which of Aidan and Binya is right. If you think that Aidan is right, calculate the value of \(\sigma\) given that \(\mathrm { P } ( \bar { X } > 32 ) = 0.1\). If you think that Binya is right, calculate the value of \(\mathrm { P } ( \bar { X } > 32 )\).

1 The random variable \(Y\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). It is found that \(\mathrm { P } ( Y > 150.0 ) = 0.0228\) and \(\mathrm { P } ( Y > 143.0 ) = 0.9332\). Find the values of \(\mu\) and \(\sigma\).

4 A petrol station finds that its daily sales, in litres, are normally distributed with mean 4520 and standard deviation 560.

1. Find on how many days of the year (365 days) the daily sales can be expected to exceed 3900 litres.
   The daily sales at another petrol station are \(X\) litres, where \(X\) is normally distributed with mean \(m\) and standard deviation 560. It is given that \(\mathrm { P } ( X > 8000 ) = 0.122\).
2. Find the value of \(m\).
3. Find the probability that daily sales at this petrol station exceed 8000 litres on fewer than 2 of 6 randomly chosen days.
   

6 The weight of a plastic box manufactured by a company is \(W\) grams, where \(W \sim \mathrm {~N} ( \mu , 20.25 )\). A significance test of the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 50.0\), against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 50.0\), is carried out at the \(5 \%\) significance level, based on a sample of size \(n\).

1. Given that \(n = 81\),
   1. find the critical region for the test, in terms of the sample mean \(\bar { W }\),
   2. find the probability that the test results in a Type II error when \(\mu = 50.2\).
   3. State how the probability of this Type II error would change if \(n\) were greater than 81 .

2 The random variable \(H\) has the distribution \(\mathrm { N } \left( \mu , 5 ^ { 2 } \right)\). The mean of a sample of \(n\) observations of \(H\) is denoted by \(\bar { H }\). It is given that \(\mathrm { P } ( \bar { H } > 53.28 ) = 0.0250\) and \(\mathrm { P } ( \bar { H } < 51.65 ) = 0.0968\), both correct to 4 decimal places. Find the values of \(\mu\) and \(n\).

1 The random variable \(H\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 105.0 ) = 0.2420\) and \(\mathrm { P } ( H > 110.0 ) = 0.6915\). Find the values of \(\mu\) and \(\sigma\), giving your answers to a suitable degree of accuracy.

6 The continuous random variable \(R\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The results of 100 observations of \(R\) are summarised by $$\Sigma r = 3360.0 , \quad \Sigma r ^ { 2 } = 115782.84 .$$

1. Calculate an unbiased estimate of \(\mu\) and an unbiased estimate of \(\sigma ^ { 2 }\).
2. The mean of 9 observations of \(R\) is denoted by \(\bar { R }\). Calculate an estimate of \(\mathrm { P } ( \bar { R } > 32.0 )\).
3. Explain whether you need to use the Central Limit Theorem in your answer to part (ii).

3 Tennis balls are dropped from a standard height, and the height of bounce, \(H \mathrm {~cm}\), is measured. \(H\) is a random variable with the distribution \(\mathrm { N } \left( 40 , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 32 ) = 0.2\).

1. Find the value of \(\sigma\).
2. 90 tennis balls are selected at random. Use an appropriate approximation to find the probability that more than 19 have \(H < 32\).

2 The random variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). It is given that $$\mathrm { P } ( Y < 48.0 ) = \mathrm { P } ( Y > 57.0 ) = 0.0668 .$$ Find the value \(y _ { 0 }\) such that \(\mathrm { P } \left( Y > y _ { 0 } \right) = 0.05\).

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 In an English language test for 12-year-old children, the raw scores, \(X\), are Normally distributed with mean 45.3 and standard deviation 11.5.

1. Find
   (A) \(\mathrm { P } ( X < 50 )\),
   (B) \(\mathrm { P } ( 45.3 < X < 50 )\).
2. Find the least raw score which would be obtained by the highest scoring \(10 \%\) of children.
3. The raw score is then scaled so that the scaled score is Normally distributed with mean 100 and standard deviation 15. This scaled score is then rounded to the nearest integer. Find the probability that a randomly selected child gets a rounded score of exactly 111 .
4. In a Mathematics test for 12-year-old children, the raw scores, \(Y\), are Normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(\mathrm { P } ( Y < 15 ) = 0.3\) and \(\mathrm { P } ( Y < 22 ) = 0.8\), find the values of \(\mu\) and \(\sigma\).

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.