Linear combinations of normal random variables

2 The random variable \(X\) has the distribution \(\mathrm { N } \left( 31.2,10.4 ^ { 2 } \right)\). Two independent random values of \(X\), denoted by \(X _ { 1 }\) and \(X _ { 2 }\), are chosen. Find \(\mathrm { P } \left( X _ { 1 } > 3 X _ { 2 } \right)\).

6 The random variable \(T\) denotes the time, in seconds, for 100 m races run by Tania. \(T\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). A random sample of 40 races run by Tania gave the following results. $$n = 40 \quad \Sigma t = 560 \quad \Sigma t ^ { 2 } = 7850$$

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
   The random variable \(S\) denotes the time, in seconds, for 100 m races run by Suki. \(S\) has the independent distribution \(\mathrm { N } ( 14.2,0.3 )\).
2. Using your answers to part (a), find the probability that, in a randomly chosen 100 m race, Suki's time will be at least 0.1 s more than Tania's time.

4 Th lifetimes, in b s, b Lg ie lig b b ad Ee rlw lig b b be tb id pd n id strib in \(\mathrm { N } \left( \mathrm { LS } ^ { 2 } \right)\) adN ( \(\mathrm { L } ^ { 2 }\) ) resp ctie ly.

1. Fid th pb b lity th t to to al 6 th lifetimes 6 fie rach ly cb en \(L \mathbf { b }\) ie \(\mathbf { b }\) b is less th \(\mathrm { HB } \quad \mathrm { Ch } \quad \mathrm { S }\).
   [0pt] [4]
2. Fid th pb b lity th tth lifetime 6 a rach lycb en En rlw b b is at least th ee times th \(t\) 6 a rach lyc b erL b ir b b

2 The independent random variables \(X\) and \(Y\) have the distributions \(\mathrm { N } ( 6.5,14 )\) and \(\mathrm { N } ( 7.4,15 )\) respectively. Find \(\mathrm { P } ( 3 X - Y < 20 )\).

1 The number, \(x\), of pine trees was counted in each of 40 randomly chosen regions of equal size in country \(A\). The number, \(y\), of pine trees was counted in each of 60 randomly chosen regions of the same equal size in country \(B\). The results are summarised as follows. $$\sum x = 752 \quad \sum x ^ { 2 } = 14320 \quad \sum y = 1548 \quad \sum y ^ { 2 } = 40200$$ Find a 95\% confidence interval for the difference between the mean number of pine trees in regions of this size in countries \(A\) and \(B\).

1 The independent random variables \(X\) and \(Y\) have standard deviations 3 and 6 respectively. Calculate the standard deviation of \(4 X - 5 Y\).

3 The lengths, in centimetres, of two types of insect, \(A\) and \(B\), are modelled by the random variables \(X \sim \mathrm {~N} ( 6.2,0.36 )\) and \(Y \sim \mathrm {~N} ( 2.4,0.25 )\) respectively. Find the probability that the length of a randomly chosen type \(A\) insect is greater than the sum of the lengths of 3 randomly chosen type \(B\) insects.

2 An examination consists of a written paper and a practical test. The written paper marks ( \(M\) ) have mean 54.8 and standard deviation 16.0. The practical test marks ( \(P\) ) are independent of the written paper marks and have mean 82.4 and standard deviation 4.8. The final mark is found by adding \(75 \%\) of \(M\) to \(25 \%\) of \(P\). Find the mean and standard deviation of the final marks for the examination. [3]

3 Sugar and flour for making cakes are measured in cups. The mass, in grams, of one cup of sugar has the distribution \(\mathrm { N } ( 250,10 )\). The mass, in grams, of one cup of flour has the independent distribution \(\mathrm { N } ( 160,9 )\). Each cake contains 2 cups of sugar and 5 cups of flour. Find the probability that the total mass of sugar and flour in one cake exceeds 1310 grams.

3 Weights of garden tables are normally distributed with mean 36 kg and standard deviation 1.6 kg . Weights of garden chairs are normally distributed with mean 7.3 kg and standard deviation 0.4 kg . Find the probability that the total weight of 2 randomly chosen tables is more than the total weight of 10 randomly chosen chairs.

3 The mass, in tonnes, of iron ore produced per day at a mine is normally distributed with mean 7.0 and standard deviation 0.46. Find the probability that the total amount of iron ore produced in 10 randomly chosen days is more than 71 tonnes.

6 The independent random variables \(X\) and \(Y\) have normal distributions with the same variance \(\sigma ^ { 2 }\). Samples of 5 observations of \(X\) and 10 observations of \(Y\) are made, and the results are summarised by \(\Sigma x = 15 , \Sigma x ^ { 2 } = 128 , \Sigma y = 36\) and \(\Sigma y ^ { 2 } = 980\). Find a pooled estimate of \(\sigma ^ { 2 }\).

4 The masses of a certain variety of potato are normally distributed with mean 180 g and variance \(1550 \mathrm {~g} ^ { 2 }\). Two potatoes of this variety are chosen at random. Find the probability that the mass of one of these potatoes is at least twice the mass of the other.

4 A certain make of washing machine has a wash-time with mean 56.9 minutes and standard deviation 4.8 minutes. A certain make of tumble dryer has a drying-time with mean 61.1 minutes and standard deviation 6.3 minutes. Both times are normally distributed and are independent of each other. Find the probability that a randomly chosen wash-time differs by more than 3 minutes from a randomly chosen drying-time.

1 A random variable \(X\) has the distribution \(\mathrm { N } ( 410,400 )\).
Find the probability that the mean of a random sample of 36 values of \(X\) is less than 405 .

2 A random sample of \(n\) people were questioned about their internet use. 87 of them had a high-speed internet connection. A confidence interval for the population proportion having a high-speed internet connection is \(0.1129 < p < 0.1771\).

1. Write down the mid-point of this confidence interval and hence find the value of \(n\).
2. This interval is an \(\alpha \%\) confidence interval. Find \(\alpha\).

1 The lengths of logs are normally distributed with mean 3.5 m and standard deviation 0.12 m . Describe fully the distribution of the total length of 8 randomly chosen logs.

1. The continuous random variable \(X\) has the distribution \(\mathrm { N } ( \mu , 30 )\). The mean of a random sample of 8 observations of \(X\) is 53.1. Determine a \(95 \%\) confidence interval for \(\mu\). You should give the end points of the interval correct to 4 significant figures.

4 A factory supplies boxes of children's bricks. Each box contains 10 randomly chosen large bricks and 20 randomly chosen small bricks. The masses, in grams, of large and small bricks have the distributions \(\mathrm { N } ( 60,1.2 )\) and \(\mathrm { N } ( 30,0.7 )\) respectively. The mass of an empty box is 8 g . Find the probability that the total weight of a box and its contents is less than 1200 g .

7 Bags of sugar are packed in boxes, each box containing 20 bags. The masses of the boxes, when empty, are normally distributed with mean 0.4 kg and standard deviation 0.01 kg . The masses of the bags are normally distributed with mean 1.02 kg and standard deviation 0.03 kg .

1. Find the probability that the total mass of a full box of 20 bags is less than 20.6 kg .
2. Two full boxes are chosen at random. Find the probability that they differ in mass by less than 0.02 kg .

2 The lengths of sewing needles in travel sewing kits are distributed normally with mean \(\mu \mathrm { mm }\) and standard deviation 1.5 mm . A random sample of \(n\) needles is taken. Find the smallest value of \(n\) such that the width of a \(95 \%\) confidence interval for the population mean is at most 1 mm .

1 Exam marks, \(X\), have mean 70 and standard deviation 8.7. The marks need to be scaled using the formula \(Y = a X + b\) so that the scaled marks, \(Y\), have mean 55 and standard deviation 6.96. Find the values of \(a\) and \(b\).

2 The random variable \(X\) has the distribution \(\mathrm { N } ( 3,1.2 )\). The random variable \(A\) is defined by \(A = 2 X\). The random variable \(B\) is defined by \(B = X _ { 1 } + X _ { 2 }\), where \(X _ { 1 }\) and \(X _ { 2 }\) are independent random values of \(X\). Describe fully the distribution of \(A\) and the distribution of \(B\). Distribution of \(A\) : \(\_\_\_\_\)
Distribution of \(B\) : \(\_\_\_\_\)

5 The times, in months, taken by a builder to build two types of house, \(P\) and \(Q\), are represented by the independent variables \(T _ { 1 } \sim \mathrm {~N} \left( 2.2,0.4 ^ { 2 } \right)\) and \(T _ { 2 } \sim \mathrm {~N} \left( 2.8,0.5 ^ { 2 } \right)\) respectively.

1. Find the probability that the total time taken to build one house of each type is less than 6 months.
2. Find the probability that the time taken to build a type \(Q\) house is more than 1.2 times the time taken to build a type \(P\) house.

3 The times, in minutes, taken by people to complete a walk are normally distributed with mean \(\mu\). The times, \(t\) minutes, for a random sample of 80 people were summarised as follows. $$\Sigma t = 7220 \quad \Sigma t ^ { 2 } = 656060$$

1. Calculate a \(97 \%\) confidence interval for \(\mu\).
2. Explain whether it was necessary to use the Central Limit theorem in part (i).

6 The times, in minutes, taken to complete the two parts of a task are normally distributed with means 4.5 and 2.3 respectively and standard deviations 1.1 and 0.7 respectively.

1. Find the probability that the total time taken for the task is less than 8.5 minutes.
2. Find the probability that the time taken for the first part of the task is more than twice the time taken for the second part.

2 Two brands of car battery, ‘Invincible’ and ‘Excelsior’, have lifetimes which are normally distributed. Invincible batteries have a mean lifetime of 5 years with standard deviation 0.7 years. Excelsior batteries have a mean lifetime of 4.5 years with standard deviation 0.5 years. Random samples of 20 Invincible batteries and 25 Excelsior batteries are selected and the sample mean lifetimes are \(\bar { X } _ { I }\) years and \(\bar { X } _ { E }\) years respectively.

1. State the distributions of \(\bar { X } _ { I }\) and \(\bar { X } _ { E }\).
2. Calculate \(\mathrm { P } \left( \bar { X } _ { I } - \bar { X } _ { E } \geqslant 1 \right)\).

1 The random variable \(X\) has the distribution \(\mathrm { B } ( 10,0.15 )\). Find the probability that the mean of a random sample of 50 observations of \(X\) is greater than 1.4.

1 A random variable \(X\) has the distribution \(\mathrm { N } ( 410,400 )\).
Find the probability that the mean of a random sample of 36 values of \(X\) is less than 405.

1 An analysis of a random sample of 150 urban dwellings for sale showed that 102 are semi-detached. An analysis of an independent random sample of 80 rural dwellings for sale showed that 36 are semi-detached.

1. Construct an approximate \(99 \%\) confidence interval for the difference between the proportion of urban dwellings for sale that are semi-detached and the proportion of rural dwellings for sale that are semi-detached.
2. Hence comment on the claim that there is no difference between these two proportions.

6 The mean Intelligence Quotient (IQ) of a random sample of 15 pupils at \(\operatorname { School } A\) is 109 . The mean IQ of a random sample of 20 pupils at School \(B\) is 112 . You may assume that the IQs for the populations from which these samples are taken are normally distributed, and that both distributions have standard deviation 15. Find a \(90 \%\) confidence interval for \(\mu _ { B } - \mu _ { A }\), where \(\mu _ { A }\) and \(\mu _ { B }\) are the population mean IQs.

1 The blood-test procedure at a clinic is that a person arrives, takes a numbered ticket and waits for that number to be called. The waiting times between the numbers called have independent normal distributions with mean 3.5 minutes and standard deviation 0.9 minutes. My ticket is number 39 and as I take my ticket number 1 is being called, so that I have to wait for 38 numbers to be called. Find the probability that I will have to wait between 120 minutes and 140 minutes.

4 The volume, in millilitres, of a small cup of coffee has the distribution \(\mathrm { N } ( 103.4,10.2 )\). The volume of a large cup of coffee is 1.5 times the volume of a small cup of coffee.

1. Find the mean and standard deviation of the volume of a large cup of coffee.
2. Find the probability that the total volume of a randomly chosen small cup of coffee and a randomly chosen large cup of coffee is greater than 250 ml .

6 A factory makes loaves of bread in batches. One batch of loaves contains \(X\) kilograms of dried yeast and \(Y\) kilograms of flour, where \(X\) and \(Y\) have the independent distributions \(\mathrm { N } \left( 0.7,0.02 ^ { 2 } \right)\) and \(\mathrm { N } \left( 100.0,3.0 ^ { 2 } \right)\) respectively. Dried yeast costs \(
) 13.50\( per kilogram and flour costs \)\\( 0.90\) per kilogram. For making one batch of bread the total of all other costs is \(
) 55\(. The factory sells each batch of bread for \)\\( 200\). Find the probability that the profit made on one randomly chosen batch of bread is greater than \(
) 40$. [7]

4 The mass, in tonnes, of steel produced per day at a factory is normally distributed with mean 65.2 and standard deviation 3.6. It can be assumed that the mass of steel produced each day is independent of other days. The factory makes \(
) 50$ profit on each tonne of steel produced. Find the probability that the total profit made in a randomly chosen 7-day week is less than \(
) 22000$.

1. A machine puts liquid into bottles of perfume. The amount of liquid put into each bottle, \(D \mathrm { ml }\), follows a normal distribution with mean 25 ml

Given that 15\% of bottles contain less than 24.63 ml

1. find, to 2 decimal places, the value of \(k\) such that \(\mathrm { P } ( 24.63 < D < k ) = 0.45\) A random sample of 200 bottles is taken.
2. Using a normal approximation, find the probability that fewer than half of these bottles contain between 24.63 ml and \(k \mathrm { ml }\) The machine is adjusted so that the standard deviation of the liquid put in the bottles is now 0.16 ml Following the adjustments, Hannah believes that the mean amount of liquid put in each bottle is less than 25 ml She takes a random sample of 20 bottles and finds the mean amount of liquid to be 24.94 ml
3. Test Hannah's belief at the \(5 \%\) level of significance. You should state your hypotheses clearly.

7 A random sample of 8 swimmers from a swimming club were timed over a distance of 100 metres, once in an outdoor pool and once in an indoor pool. Their times, in seconds, are given in the following table. Assuming a normal distribution, test, at the \(5 \%\) significance level, whether there is a non-zero difference between mean time in the outdoor pool and mean time in the indoor pool.