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

6. A potter makes decorative tiles in two colours, red and yellow. The length, \(R \mathrm {~cm}\), of the red tiles has a normal distribution with mean 15 cm and standard deviation 1.5 cm . The length, \(Y \mathrm {~cm}\), of the yellow tiles has the normal distribution \(\mathrm { N } \left( 12,0.8 ^ { 2 } \right)\). The random variables \(R\) and \(Y\) are independent. A red tile and a yellow tile are chosen at random.

1. Find the probability that the yellow tile is longer than the red tile. Taruni buys 3 red tiles and 1 yellow tile.
2. Find the probability that the total length of the 3 red tiles is less than 4 times the length of the yellow tile. Stefan defines the random variable \(X = a R + b Y\), where \(a\) and \(b\) are constants. He wants to use values of \(a\) and \(b\) such that \(X\) has a mean of 780 and minimum variance.
3. Find the value of \(a\) and the value of \(b\) that Stefan should use. \includegraphics[max width=\textwidth, alt={}, center]{ba3f3f9c-53d2-4e95-b2f3-3f617f1821ed-19_2255_50_314_34}

2. Krishi owns a farm on which he keeps chickens. He selects, at random, 10 of the eggs produced and weighs each of them.
You may assume that these weights are a random sample from a normal distribution with standard deviation 1.9 g The total weight of these 10 eggs is 537.2 g

1. Find a \(95 \%\) confidence interval for the mean weight of the eggs produced by Krishi's chickens. Krishi was hoping to obtain a \(99 \%\) confidence interval of width at most 1.5 g
2. Calculate the minimum sample size necessary to achieve this. \includegraphics[max width=\textwidth, alt={}, center]{fc43aabf-ad04-4852-8539-981cef608f31-04_2662_95_107_1962}

1. A market stall sells vegetables. Two of the vegetables sold are broccoli heads and cabbages.

The weights of these broccoli heads, \(B\) kilograms, follow a normal distribution $$B \sim \mathrm {~N} \left( 0.588,0.084 ^ { 2 } \right)$$ The weights of these cabbages, \(C\) kilograms, follow a normal distribution $$C \sim \mathrm {~N} \left( 0.908,0.039 ^ { 2 } \right)$$

1. Find the probability that the total weight of two randomly chosen broccoli heads is less than the weight of a randomly chosen cabbage. Broccoli heads cost \(\pounds 2.50\) per kg and cabbages cost \(\pounds 3.00\) per kg. Jaymini buys 1 broccoli head and 2 cabbages, chosen randomly.
2. Find the probability that she pays more than £7 The market stall offers a discount for buying 5 or more broccoli heads. The price with the discount is \(\pounds w\) per kg. Let \(\pounds D\) be the price with the discount of 5 broccoli heads.
3. Find, in terms of \(w\), the mean and standard deviation of \(D\) Given that \(\mathrm { P } ( D < 6 ) < 0.1\)
4. find the smallest possible value of \(w\), giving your answer to 2 decimal places.

1. The weights, \(x \mathrm {~kg}\), of each of 10 watermelons selected at random from Priya's shop were recorded. The results are summarised as follows

$$\sum x = 114.2 \quad \sum x ^ { 2 } = 1310.464$$

1. Calculate unbiased estimates of the mean and the variance of the weights of the watermelons in Priya's shop. Priya researches the weight of watermelons, for the variety she has in her shop, and discovers that the weights of these watermelons are normally distributed with a standard deviation of 0.8 kg
2. Calculate a \(95 \%\) confidence interval for the mean weight of watermelons in Priya's shop. Give the limits of your confidence interval to 2 decimal places. Priya claims that the confidence interval in part (b) suggests that nearly all of the watermelons in her shop weigh more than 10.5 kg
3. Use your answer to part (b) to estimate the smallest proportion of watermelons in her shop that weigh less than 10.5 kg

1. Charlie is training for three events: a 1500 m swim, a 40 km bike ride and a 10 km run.

From past experience his times, in minutes, for each of the three events independently have the following distributions. $$\begin{aligned} & S \sim \mathrm {~N} \left( 41,5.2 ^ { 2 } \right) \text { represents the time for the swim } \\ & B \sim \mathrm {~N} \left( 81,4.2 ^ { 2 } \right) \text { represents the time for the bike ride } \\ & R \sim \mathrm {~N} \left( 57,6.6 ^ { 2 } \right) \text { represents the time for the run } \end{aligned}$$

1. Find the probability that Charlie's total time for a randomly selected swim, bike ride and run exceeds 3 hours.
2. Find the probability that the time for a randomly selected swim will be at least 20 minutes quicker than the time for a randomly selected run. Given that \(\mathrm { P } ( S + B + R > t ) = 0.95\)
3. find the value of \(t\) A triathlon consists of a 1500 m swim, immediately followed by a 40 km bike ride, immediately followed by a 10 km run. Charlie uses the answer to part (a) to find the probability that, in 6 successive independent triathlons, his time will exceed 3 hours on at least one occasion.
4. Find the answer Charlie should obtain. Jane says that Charlie should not have used the answer to part (a) for the calculation in part (d).
5. Explain whether or not Jane is correct.

1. A farmer sells strawberries in baskets. The contents of each of 100 randomly selected baskets were weighed and the results, given to the nearest gram, are shown below.

The farmer proposes that the weight of strawberries per basket, in grams, should be modelled by a normal distribution with a mean of 310 g and standard deviation 4 g . Using his model, the farmer obtains the following expected frequencies.

1. Find the value of \(a\) and the value of \(b\). Give your answers correct to one decimal place. Before \(s \leqslant 303.5\) and \(s > 315.5\) are included, for the remaining cells, $$\sum \frac { ( O - E ) ^ { 2 } } { E } = 9.71$$
2. Using a 5\% significance level, test whether the data are consistent with the model. You should state your hypotheses, the test statistic and the critical value used. An alternative model uses estimates for the population mean and standard deviation from the data given. Using these estimated values no expected frequency is below 5
   Another test is to be carried out, using a \(5 \%\) significance level, to assess whether the data are consistent with this alternative model.
3. State the effect, if any, on the critical value for this test. Give a reason for your answer.

1 A machine fills bottles with mineral water.
The machine is checked every day to ensure that it is working correctly. On a particular day a random sample of 100 bottles is taken. The volume of water, \(x\) millilitres, for each bottle is measured and each measurement is coded using $$y = x - 1000$$ The results are summarised below $$\sum y = 847 \quad \sum y ^ { 2 } = 13510.09$$

1. 1. Show that the value of the unbiased estimate of the mean of \(x\) is 1008.47
   2. Calculate the unbiased estimate of the variance of \(x\) The machine was initially set so that the volume of water in a bottle had a mean value of 1010 millilitres. Later, a test at the \(5 \%\) significance level is used to determine whether or not the mean volume of water in a bottle has changed. If it has changed then the machine is stopped and reset.
2. Write down suitable null and alternative hypotheses for a 2-tailed test.
3. Find the critical region for \(\bar { X }\) in the above test.
4. Using your answer to part (a) and your critical region found in part (c), comment on whether or not the machine needs to be stopped and reset.
   Give a reason for your answer.
5. Explain why the use of \(\sigma ^ { 2 } = s ^ { 2 }\) is reasonable in this situation.

6 A garden centre sells bags of stones and large bags of gravel.
The weight, \(X\) kilograms, of stones in a bag can be modelled by a normal distribution with unknown mean \(\mu\) and known standard deviation 0.4 The stones in each of a random sample of 36 bags from a large batch is weighed. The total weight of stones in these 36 bags is found to be 806.4 kg

1. Find a 98\% confidence interval for the mean weight of stones in the batch.
2. Explain why the use of the Central Limit theorem is not required to answer part (a) The manufacturer of these bags of stones claims that bags in this batch have a mean weight of 22.5 kg
3. Using your answer to part (a), comment on the claim made by the manufacturer. The weight, \(Y\) kilograms, of gravel in a large bag can be modelled by a normal distribution with mean 850 kg and standard deviation 5 kg A builder purchases 10 large bags of gravel.
4. Find the probability that the mean weight of gravel in the 10 large bags is less than 848 kg

7 At a particular supermarket, the times taken to serve each customer in a queue at a standard checkout may be modelled by a normal distribution with mean 240 seconds and standard deviation 20 seconds. There is a queue of 3 customers at a standard checkout.
Making a reasonable assumption about the times taken to serve these customers,

1. find the probability that the total time taken to serve the 3 customers will be less than 11 minutes.
2. State the assumption you have made in part (a) In the supermarket there is also an express checkout, which is reserved for customers buying 10 or fewer items. The time taken to serve a customer at this express checkout may be modelled by a normal distribution with mean 100 seconds and standard deviation 8 seconds. On a particular day Jiang has 8 items to pay for and has to choose whether to join a queue of 3 customers waiting at a standard checkout or a queue of 7 customers waiting at the express checkout. Using a similar assumption to that made in part (a),
3. find the probability that the total time taken to serve the 3 customers at the standard checkout will exceed the total time taken to serve the 7 customers at the express checkout.

1. Small containers and large containers are independently filled with fruit juice.

The amounts of fruit juice in small containers are normally distributed with mean 180 ml and standard deviation 4.5 ml The amounts of fruit juice in large containers are normally distributed with mean 330 ml and standard deviation 6.7 ml The random variable \(W\) represents the total amount of fruit juice in a random sample of 2 small containers minus the amount of fruit juice in 1 randomly selected large container. \(W \sim \mathrm {~N} ( a , b )\) where \(a\) and \(b\) are positive constants.

1. Find the value of \(a\) and the value of \(b\)
2. Find the probability that a randomly chosen large container of fruit juice contains more than 1.8 times the amount of fruit juice in a randomly chosen small container. A random sample of 3 small containers of fruit juice is taken.
3. Find the probability that the first container of fruit juice in this sample contains at least 5 ml more than the mean amount of fruit juice in all 3 small containers.

6. A company produces a certain type of mug. The masses of these mugs are normally distributed with mean \(\mu\) and standard deviation 1.2 grams. A random sample of 5 mugs is taken and the mass, in grams, of each mug is measured. The results are given below. \section*{\(\begin{array} { l l l l l } 229.1 & 229.6 & 230.9 & 231.2 & 231.7 \end{array}\)}

1. Find a \(95 \%\) confidence interval for \(\mu\), giving your limits correct to 1 decimal place. Sonia plans to take 20 random samples, each of 5 mugs. A 95\% confidence interval for \(\mu\) is to be determined for each sample.
2. Find the probability that more than 3 of these intervals will not contain \(\mu\).

7. The independent random variables \(X\) and \(Y\) are such that $$X \sim \mathrm {~N} \left( 30,4.5 ^ { 2 } \right) \text { and } Y \sim \mathrm {~N} \left( 20,3.5 ^ { 2 } \right)$$ The random variables \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) are independent and each has the same distribution as \(X\). The random variables \(Y _ { 1 }\) and \(Y _ { 2 }\) are independent and each has the same distribution as \(Y\). Given that the random variable \(A\) is defined as $$A = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } + Y _ { 1 } + Y _ { 2 } } { 5 }$$

1. find \(\mathrm { P } ( A < 24 )\) The random variable \(W\) is such that \(W \sim \mathrm {~N} \left( \mu , 2.8 ^ { 2 } \right)\) Given that \(\mathrm { P } ( W - X < 4 ) = 0.1\) and that \(W\) and \(X\) are independent,
2. find the value of \(\mu\), giving your answer to 3 significant figures.

5. A factory produces steel sheets whose weights, \(X \mathrm {~kg}\), have a normal distribution with an unknown mean \(\mu \mathrm { kg }\) and known standard deviation \(\sigma \mathrm { kg }\). A random sample of 25 sheets gave both a

• \(95 \%\) confidence interval for \(\mu\) of \(( 30.612,31.788 )\)
• \(c \%\) confidence interval for \(\mu\) of \(( 30.66,31.74 )\)
  1. Find the value of \(\sigma\)
  2. Find the value of \(c\), giving your answer correct to 3 significant figures.

7.(i)As part of a recruitment exercise candidates are required to complete three separate tasks.The times taken,\(A , B\) and \(C\) ,in minutes,for candidates to complete the three tasks are such that $$A \sim \mathrm {~N} \left( 21,2 ^ { 2 } \right) , B \sim \mathrm {~N} \left( 32,7 ^ { 2 } \right) \text { and } C \sim \mathrm {~N} \left( 45,9 ^ { 2 } \right)$$ The time taken by an individual candidate to complete each task is assumed to be independent of the time taken to complete each of the other tasks. A candidate is selected at random.

1. Find the probability that the candidate takes a total time of more than 90 minutes to complete all three tasks.
2. Find \(\mathrm { P } ( A > B )\) (ii)A simple random sample,\(X _ { 1 } , X _ { 2 } , X _ { 3 } , X _ { 4 }\) ,is taken from a normal population with mean \(\mu\) and standard deviation \(\sigma\) Given that $$\bar { X } = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } + X _ { 4 } } { 4 }$$ and that $$\mathrm { P } \left( X _ { 1 } > \bar { X } + k \sigma \right) = 0.1$$ where \(k\) is a constant,
   find the value of \(k\) ,giving your answer correct to 3 significant figures.

   END

1. Components are manufactured such that their length in mm is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). Below is a 95\% confidence interval for \(\mu\) calculated from a random sample of components.
   (11.52, 13.75)

Using the same random sample,

1. find a \(90 \%\) confidence interval for \(\mu\). Four 90\% confidence intervals are found from independent random samples.
2. Calculate the probability that only 3 of these 4 intervals will contain \(\mu\).

1. A baker produces bread buns and bread rolls. The weights of buns, \(B\) grams, and the weights of rolls, \(R\) grams, are such that \(B \sim \mathrm {~N} \left( 55,1.3 ^ { 2 } \right)\) and \(R \sim \mathrm {~N} \left( 51,1.2 ^ { 2 } \right)\)

A bun and a roll are selected at random.

1. Find the probability that the bun weighs less than \(110 \%\) of the weight of the roll. Two buns are chosen at random.
2. Find the probability that their weights differ by more than 1 gram. The baker sells bread in bags. Each bag contains either 10 buns or 11 rolls. The weight of an empty bag, \(S\) grams, is such that \(S \sim \mathrm {~N} \left( 3,0.2 ^ { 2 } \right)\)
3. Find the probability that a bag of buns weighs less than a bag of rolls.

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

A random sample of 25 observations of \(X\) produced a \(95 \%\) confidence interval for \(\mu\) of (26.624, 28.976)

1. Find the mean of the sample.
2. Show that the standard deviation is 3 The \(a\) \% confidence interval using the 25 observations has a width of 2.1
3. Calculate the value of \(a\)
4. Find the smallest sample size, of observations from \(X\), that would be required to obtain a 95\% confidence interval of width at most 1.5

6 A particular lift has a maximum load capacity of 700 kg .
The weights of men are normally distributed with mean 80 kg and standard deviation 10 kg . The weights of women are normally distributed with mean 69 kg and standard deviation 5 kg . You may assume that weights of people are independent.

1. Find the probability that when 6 men and 3 women are in the lift, the load exceeds 700 kg . A sign in the lift states: "Maximum number of people in the lift is \(c\) "
2. Find the value of \(c\) such that the probability of the load exceeding 700 kg is less than \(2.5 \%\) no matter the gender of the occupants.

1. A machine makes screws with a mean length of 30 mm and a standard deviation of 2.5 mm .

A manager claims that, following some repairs, the machine is now making screws with a mean length of less than 30 mm . The manager takes a random sample of 80 screws and finds that they have a mean length of 29.5 mm . Use a suitable test, at the \(5 \%\) level of significance, to determine whether there is evidence to support the manager's claim. State your hypotheses clearly.

1. A company produces bricks.

The weight of a brick, \(B \mathrm {~kg}\), is such that \(B \sim \mathrm {~N} \left( 1.96 , \sqrt { 0.003 } ^ { 2 } \right)\) Two bricks are chosen at random.

1. Find the probability that the difference in weight of the 2 bricks is greater than 0.1 kg A random sample of \(n\) bricks is to be taken.
2. Find the minimum sample size such that the probability of the sample mean being greater than 2 is less than 1\% The bricks are randomly selected and stacked on pallets.
   The weight of an empty pallet, \(E \mathrm {~kg}\), is such that \(E \sim \mathrm {~N} \left( 21.8 , \sqrt { 0.6 } ^ { 2 } \right)\) The random variable \(M\) represents the total weight of a pallet stacked with 500 bricks. The random variable \(T\) represents the total weight of a container of cement.
   Given that \(T\) is independent of \(M\) and that \(T \sim \mathrm {~N} \left( 774 , \sqrt { 1.8 } ^ { 2 } \right)\)
3. calculate \(\mathrm { P } ( 4 T > 100 + 3 M )\)

4. A farm produces potatoes. The potatoes are packed into sacks. The weight of a sack of potatoes is modelled by a normal distribution with mean 25.6 kg and standard deviation 0.24 kg

1. Find the probability that two randomly chosen sacks of potatoes differ in weight by more than 0.5 kg Sacks of potatoes are randomly selected and packed onto pallets. The weight of an empty pallet is modelled by a normal distribution with mean 20.0 kg and standard deviation 0.32 kg Each full pallet of potatoes holds 30 sacks of potatoes.
2. Find the probability that the total weight of a randomly chosen full pallet of potatoes is greater than 785 kg
   
   \includegraphics[max width=\textwidth, alt={}, center]{0434a6c1-686a-449d-ba16-dbb8e60288e8-15_2258_51_313_36}

2. Philip and James are racing car drivers. Philip's lap times, in seconds, are normally distributed with mean 90 and variance 9. James' lap times, in seconds, are normally distributed with mean 91 and variance 12. The lap times of Philip and James are independent. Before a race, they each take a qualifying lap.

1. Find the probability that James' time for the qualifying lap is less than Philip's. The race is made up of 60 laps. Assuming that they both start from the same starting line and lap times are independent,
2. find the probability that Philip beats James in the race by more than 2 minutes.

3. A woodwork teacher measures the width, \(w \mathrm {~mm}\), of a board. The measured width, \(X \mathrm {~mm}\), is normally distributed with mean \(w \mathrm {~mm}\) and standard deviation 0.5 mm .

1. Find the probability that \(X\) is within 0.6 mm of \(w\). The same board is measured 16 times and the results are recorded.
2. Find the probability that the mean of these results is within 0.3 mm of \(w\). Given that the mean of these 16 measurements is 35.6 mm ,
3. find a 98\% confidence interval for \(w\).

2. A workshop makes two types of electrical resistor. The resistance, \(X\) ohms, of resistors of Type A is such that \(X \sim \mathrm {~N} ( 20,4 )\).
The resistance, \(Y\) ohms, of resistors of Type B is such that \(Y \sim \mathrm {~N} ( 10,0.84 )\).
When a resistor of each type is connected into a circuit, the resistance \(R\) ohms of the circuit is given by \(R = X + Y\) where \(X\) and \(Y\) are independent. Find

1. \(\mathrm { E } ( R )\),
2. \(\operatorname { Var } ( R )\),
3. \(\mathrm { P } ( 28.9 < R < 32.64 )\) (6)

7. A set of scaffolding poles come in two sizes, long and short. The length \(L\) of a long pole has the normal distribution \(\mathrm { N } \left( 19.7,0.5 ^ { 2 } \right)\). The length \(S\) of a short pole has the normal distribution \(\mathrm { N } \left( 4.9,0.2 ^ { 2 } \right)\). The random variables \(L\) and \(S\) are independent. A long pole and a short pole are selected at random.

1. Find the probability that the length of the long pole is more than 4 times the length of the short pole. Four short poles are selected at random and placed end to end in a row. The random variable \(T\) represents the length of the row.
2. Find the distribution of \(T\).
3. Find \(\mathrm { P } ( | L - T | < 0.1 )\).