2.04e Normal distribution: as model N(mu, sigma^2)

The weights of packages delivered to Susie are normally distributed with a mean of 510 grams and a standard deviation of 45 grams.

1. Find the probability that a randomly selected package delivered to Susie weighs less than 450 grams. The heaviest 5\% of packages delivered to Susie are delivered by Rav in his van, the others are delivered by Taruni on foot.
2. Find the weight of the lightest package that Rav would deliver to Susie. Susie randomly selects a package from those delivered by Taruni.
3. Find the probability that this package weighs more than 450 grams. On Tuesday there are 5 packages delivered to Susie.
4. Find the probability that 4 are delivered by Taruni and 1 is delivered by Rav.

3. The distance achieved in a long jump competition by students at a school is normally Students who achieve a distance greater than 4.3 metres receive a medal.

1. Find the proportion of students who receive medals. The school wishes to give a certificate of achievement or a medal to the \(80 \%\) of students who achieve a distance of at least \(d\) metres.
2. Find the value of \(d\). Of those who received medals, the \(\frac { 1 } { 3 }\) who jump the furthest will receive gold medals.
3. Find the shortest distance, \(g\) metres, that must be achieved to receive a gold medal. A journalist from the local newspaper interviews a randomly selected group of 3 medal winners.
4. Find the exact probability that there is at least one gold medal winner in the group. \includegraphics[max width=\textwidth, alt={}, center]{81d5e460-9559-4d25-aa08-6440559aec83-08_79_1153_233_251} Students who achieve a distance greater than 4.3 metres receive a medal.
   1. Find the proportion of students who receive medals.

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6. The random variable \(A\) represents the score when a spinner is spun. The probability distribution for \(A\) is given in the following table.

1. Show that \(\mathrm { E } ( A ) = 3.5\)
2. Find \(\operatorname { Var } ( A )\) The random variable \(B\) represents the score on a 4 -sided die. The probability distribution for \(B\) is given in the following table where \(k\) is a positive integer.

   \(b\)	1	3	4	\(k\)
   \(\mathrm { P } ( B = b )\)	0.25	0.25	0.25	0.25

3. Write down the name of the probability distribution of \(B\).
4. Given that \(\mathrm { E } ( B ) = \mathrm { E } ( A )\) state, giving a reason, the value of \(k\). The random variable \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) Sam and Tim are playing a game with the spinner and the die. They each spin the spinner once to obtain their value of \(A\) and each roll the die once to obtain their value of \(B\).
   Their value of \(A\) is taken as their value of \(\mu\) and their value of \(B\) is taken as their value of \(\sigma\). The person with the larger value of \(\mathrm { P } ( X > 3.5 )\) is the winner.
5. Given that Sam obtained values of \(a = 4\) and \(b = 3\) and Tim obtained \(b = 4\) find, giving a reason, the probability that Tim wins.
6. Find the largest value of \(\mathrm { P } ( X > 3.5 )\) achievable in this game.
7. Find the probability of achieving this value. \includegraphics[max width=\textwidth, alt={}, center]{81d5e460-9559-4d25-aa08-6440559aec83-21_2255_50_314_34}

Kris works in the mailroom of a large company and is responsible for all the letters sent by the company. The weights of letters sent by the company, \(W\) grams, have a normal distribution with mean 165 g and standard deviation 35 g .

1. Estimate the proportion of letters sent by the company that weigh less than 120 g . Kris splits the letters to be sent into 3 categories: heavy, medium and light, with \(\frac { 1 } { 3 }\) of the letters in each category.
2. Find the weight limits that determine medium letters. A heavy letter is chosen at random.
3. Find the probability that this letter weighs less than 200 g . Kris chooses a random sample of 3 letters from those in the mailroom one day.
4. Find the probability that there is one letter in each of the 3 categories.
   

A manufacturer fills bottles with oil. The volume of oil in a bottle, \(V \mathrm { ml }\), is normally distributed with \(V \sim \mathrm {~N} \left( 100,2.5 ^ { 2 } \right)\)

1. Find \(\mathrm { P } ( V > 104.9 )\)
2. In a pack of 150 bottles, find the expected number of bottles containing more than 104.9 ml
3. Find the value of \(v\), to 2 decimal places, such that \(\mathrm { P } ( V > v \mid V < 104.9 ) = 0.2801\)

1. A competition consists of two rounds.

The time, in minutes, taken by adults to complete round one is modelled by a normal distribution with mean 15 minutes and standard deviation 2 minutes.

1. Use standardisation to find the proportion of adults that take less than 18 minutes to complete round one. Only the fastest \(60 \%\) of adults from round one take part in round two.
2. Use standardisation to find the longest time that an adult can take to complete round one if they are to take part in round two. The time, \(T\) minutes, taken by adults to complete round two is modelled by a normal distribution with mean \(\mu\) Given that \(\mathrm { P } ( \mu - 10 < T < \mu + 10 ) = 0.95\)
3. find \(\mathrm { P } ( T > \mu - 5 \mid T > \mu - 10 )\)

1. The random variable \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\)

Given that \(\mathrm { P } ( X > \mu + a ) = 0.35\) where \(a\) is a constant, find

1. \(\mathrm { P } ( X > \mu - a )\)
2. \(\mathrm { P } ( \mu - a < X < \mu + a )\)
3. \(\mathrm { P } ( X < \mu + a \mid X > \mu - a )\)

1. A doctor is studying the scans of 30 -week old foetuses. She takes a random sample of 8 scans and measures the length, \(f \mathrm {~mm}\), of the leg bone called the femur. She obtains the following results.

$$\begin{array} { l l l l l l l l } 52 & 53 & 56 & 57 & 57 & 59 & 60 & 62 \end{array}$$

1. Show that \(\mathrm { S } _ { f f } = 80\) The doctor also measures the head circumference, \(h \mathrm {~mm}\), of each foetus and her results are summarised as $$\sum h = 2209 \quad \sum h ^ { 2 } = 610463 \quad \mathrm {~S} _ { f h } = 182$$
2. Find \(\mathrm { S } _ { h h }\)
3. Calculate the product moment correlation coefficient between the length of the femur and the head circumference for these data. The doctor believes that there is a linear relationship between the length of the femur and the head circumference of 30-week old foetuses.
4. State, giving a reason, whether or not your calculation in part (c) supports the doctor's belief.
5. Find an equation of the regression line of \(h\) on \(f\). The doctor plans in future to measure the femur length, \(f\), and then use the regression line to estimate the corresponding head circumference, \(h\). A statistician points out that there will always be the chance of an error between the true head circumference and the estimated value of the head circumference. Given that the error, \(E \mathrm {~mm}\), has the normal distribution \(\mathrm { N } \left( 0,4 ^ { 2 } \right)\)
6. find the probability that the estimate is within 3 mm of the true value.

1. The label on a jar of Amy's jam states that the jar contains about 400 grams of jam. For each jar that contains less than 388 grams of jam, Amy will be fined \(\pounds 100\). If a jar contains more than 410 grams of jam then Amy makes a loss of \(\pounds 0.30\) on that jar.

The weight of jam, \(A\) grams, in a jar of Amy's jam has a normal distribution with mean \(\mu\) grams and standard deviation \(\sigma\) grams. Amy chooses \(\mu\) and \(\sigma\) so that \(\mathrm { P } ( A < 388 ) = 0.001\) and \(\mathrm { P } ( A > 410 ) = 0.0197\)

1. Find the value of \(\mu\) and the value of \(\sigma\). Amy can sell jars of jam containing between 388 grams and 410 grams for a profit of \(\pounds 0.25\)
2. Calculate the expected amount, in £s, that Amy receives for each jar of jam.
   

1. A machine makes bolts such that the length, \(L \mathrm {~cm}\), of a bolt has distribution \(L \sim \mathrm {~N} \left( 4.1,0.125 ^ { 2 } \right)\)

A bolt is selected at random.

1. Find the probability that the length of this bolt is more than 4.3 cm .
2. Show that \(\mathrm { P } ( 3.9 < L < 4.3 )\) is 0.890 correct to 3 decimal places. The machine makes 500 bolts.
   The cost to make each bolt is 5 pence.
   Only bolts with length between 3.9 cm and 4.3 cm can be used. These are sold for 9 pence each. All the bolts that cannot be used are recycled with a scrap value of 1 pence each.
3. Calculate an estimate for the profit made on these 500 bolts. Following adjustments to the machine, the length of a bolt, \(B \mathrm {~cm}\), made by the machine is such that \(B \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) Given that \(\mathrm { P } ( B > 4.198 ) = 0.025\) and \(\mathrm { P } ( B < 4.065 ) = 0.242\)
4. find the value of \(\mu\) and the value of \(\sigma\)
5. State, giving a reason, whether the adjustments to the machine will result in a decrease or an increase in the profit made on 500 bolts.

1. The weights, \(W\) grams, of kiwi fruit grown on a farm are normally distributed with mean 80 grams and standard deviation 8 grams.

The table shows the classifications of the kiwi fruit by their weight, where \(k\) is a positive constant. One kiwi fruit is selected at random from those grown on the farm.

1. Find the probability that this kiwi fruit is Large. 35\% of the kiwi fruit are Jumbo.
2. Find the value of \(k\) to one decimal place. 75\% of Tiny kiwi fruit weigh more than \(y\) grams.
3. Find the value of \(y\) giving your answer to one decimal place.

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 \(q\) \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
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The distances travelled to work, \(D \mathrm {~km}\), by the employees at a large company are normally distributed with \(D \sim \mathrm {~N} \left( 30,8 ^ { 2 } \right)\).

1. Find the probability that a randomly selected employee has a journey to work of more than 20 km .
2. Find the upper quartile, \(Q _ { 3 }\), of \(D\).
3. Write down the lower quartile, \(Q _ { 1 }\), of \(D\). An outlier is defined as any value of \(D\) such that \(D < h\) or \(D > k\) where $$h = Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \quad \text { and } \quad k = Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
4. Find the value of \(h\) and the value of \(k\). An employee is selected at random.
5. Find the probability that the distance travelled to work by this employee is an outlier.
   

   END

2. The random variable \(X\) is normally distributed with mean 177.0 and standard deviation 6.4.

1. Find \(\mathrm { P } ( 166 < X < 185 )\). It is suggested that \(X\) might be a suitable random variable to model the height, in cm , of adult males.
2. Give two reasons why this is a sensible suggestion.
3. Explain briefly why mathematical models can help to improve our understanding of real-world problems.

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)

3. The following table shows the height \(x\), to the nearest cm , and the weight \(y\), to the nearest kg , of a random sample of 12 students.

1. On graph paper, draw a scatter diagram to represent these data.
2. Write down, with a reason, whether the correlation coefficient between \(x\) and \(y\) is positive or negative. The data in the table can be summarised as follows. $$\Sigma x = 1962 , \quad \Sigma y = 740 , \quad \Sigma y ^ { 2 } = 47746 , \quad \Sigma x y = 122783 , \quad S _ { x x } = 1745 .$$
3. Find \(S _ { x y }\). The equation of the regression line of \(y\) on \(x\) is \(y = - 106.331 + b x\).
4. Find, to 3 decimal places, the value of \(b\).
5. Find, to 3 significant figures, the mean \(\bar { y }\) and the standard deviation \(s\) of the weights of this sample of students.
6. Find the values of \(\bar { y } \pm 1.96 s\).
7. Comment on whether or not you think that the weights of these students could be modelled by a normal distribution.

7. The random variable \(X\) is normally distributed with mean 79 and variance 144 . Find

1. \(\mathrm { P } ( X < 70 )\),
2. \(\mathrm { P } ( 64 < X < 96 )\). It is known that \(\mathrm { P } ( 79 - a \leq X \leq 79 + b ) = 0.6463\). This information is shown in the figure below. \includegraphics[max width=\textwidth, alt={}, center]{df898ff4-c3ef-400c-b4f7-f4df3757941d-6_581_983_818_590} Given that \(\mathrm { P } ( X \geq 79 + b ) = 2 \mathrm { P } ( X \leq 79 - a )\),
3. show that the area of the shaded region is 0.1179 .
4. Find the value of \(b\).

5.

1. Write down two reasons for using statistical models.
2. Give an example of a random variable that could be modelled by
   1. a normal distribution,
   2. a discrete uniform distribution.

7. The heights of a group of athletes are modelled by a normal distribution with mean 180 cm and a standard deviation 5.2 cm . The weights of this group of athletes are modelled by a normal distribution with mean 85 kg and standard deviation 7.1 kg . Find the probability that a randomly chosen athlete

1. is taller than 188 cm ,
2. weighs less than 97 kg .
   (2)
3. Assuming that for these athletes height and weight are independent, find the probability that a randomly chosen athlete is taller than 188 cm and weighs more than 97 kg .
4. Comment on the assumption that height and weight are independent.

The measure of intelligence, IQ, of a group of students is assumed to be Normally distributed with mean 100 and standard deviation 15.

1. Find the probability that a student selected at random has an IQ less than 91. The probability that a randomly selected student has an IQ of at least \(100 + k\) is 0.2090 .
2. Find, to the nearest integer, the value of \(k\).
   

6. The weights of bags of popcorn are normally distributed with mean of 200 g and \(60 \%\) of all bags weighing between 190 g and 210 g .

1. Write down the median weight of the bags of popcorn.
2. Find the standard deviation of the weights of the bags of popcorn. A shopkeeper finds that customers will complain if their bag of popcorn weighs less than 180 g .
3. Find the probability that a customer will complain.

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 \(\mathrm { P } ( d < X < e )\).

The weight, \(X\) grams, of soup put in a tin by machine \(A\) is normally distributed with a mean of 160 g and a standard deviation of 5 g .
A tin is selected at random.

1. Find the probability that this tin contains more than 168 g . The weight stated on the tin is \(w\) grams.
2. Find \(w\) such that \(\mathrm { P } ( X < w ) = 0.01\) The weight, \(Y\) grams, of soup put into a carton by machine \(B\) is normally distributed with mean \(\mu\) grams and standard deviation \(\sigma\) grams.
3. Given that \(\mathrm { P } ( Y < 160 ) = 0.99\) and \(\mathrm { P } ( Y > 152 ) = 0.90\) find the value of \(\mu\) and the value of \(\sigma\).
   

A manufacturer fills jars with coffee. The weight of coffee, \(W\) grams, in a jar can be modelled by a normal distribution with mean 232 grams and standard deviation 5 grams.

1. Find \(\mathrm { P } ( W < 224 )\).
2. Find the value of \(w\) such that \(\mathrm { P } ( 232 < W < w ) = 0.20\) Two jars of coffee are selected at random.
3. Find the probability that only one of the jars contains between 232 grams and \(w\) grams of coffee.