Question 1
The flight times, `T` minutes, between two cities can be modelled by a normal distribution with a mean of 75 minutes and a standard deviation of `sigma` minutes.
(a) Given that 2% of the flight times are longer than 82 minutes, find the value of `sigma`.
(b) Find the probability that a randomly selected flight will have a flight time of more than 80 minutes.
(c) Given that a flight between the two cities takes longer than 80 minutes, find the probability that it takes less than 82 minutes.
On a particular day, there are 64 flights scheduled between these two cities.
(d) Find the expected number of flights that will have a flight time of more than 80 minutes.
(e) Find the probability that more than 6 of the flights on this particular day will have a flight time of more than 80 minutes.
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Question 2
The random variable `X` follows a normal distribution with mean `mu` and standard deviation `sigma`.
(a) Find `P(mu - 1.5 sigma < X < mu + 1.5 sigma)`.
The avocados grown on a farm have weights, in grams, that are normally distributed with mean `mu` and standard deviation `sigma`. Avocados are categorized as small, medium, large or premium, according to their weight. The following table shows the probability an avocado grown on the farm is classified as small, medium, large or premium.
The maximum weight of a small avocado is 106.2 grams.
The minimum weight of a premium avocado is 182.6 grams.
(b) Find the value of `mu` and of `sigma`.
A supermarket purchases all the avocados from the farm that weigh more than 106.2 grams.
(c) Find the probability that an avocado chosen at random from this purchase is categorized as
(i) medium;
(ii) large;
(iii) premium.
The selling prices of the different categories of avocado at this supermarket are shown in the following table:
The supermarket pays the farm $200 for the avocados and assumes it will then sell them in exactly the same proportion as purchased from the farm.
(d) According to this model, find the minimum number of avocados that must be sold so that the net profit for the supermarket is at least $438.
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Question 3
The time it takes Suzi to drive from home to work each morning is normally distributed with a mean of 35 minutes and a standard deviation of `sigma` minutes.
On 25 % of days, it takes Suzi longer than 40 minutes to drive to work.
(a) Find the value of `sigma`.
(b) On a randomly selected day, find the probability that Suzi’s drive to work will take longer than 45 minutes.
Suzi will be late to work if it takes her longer than 45 minutes to drive to work. The time it takes to drive to work each day is independent of any other day.
Suzi will work five days next week.
(c) Find the probability that she will be late to work at least one day next week.
(d) Given that Suzi will be late to work at least one day next week, find the probability that she will be late less than three times.
Suzi will work 22 days this month. She will receive a bonus if she is on time at least 20 of those days.
So far this month, she has worked 16 days and been on time 15 of those days.
(e) Find the probability that Suzi will receive a bonus.
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Question 4
A bakery makes two types of muffins: chocolate muffins and banana muffins.
The weights, `C` grams, of the chocolate muffins are normally distributed with a mean of 62g and standard deviation of 2.9g.
(a) Find the probability that a randomly selected chocolate muffin weighs less than 61g.
(b) In a random selection of 12 chocolate muffins, find the probability that exactly 5 weigh less than 61g.
The weights, `B` grams, of the banana muffins are normally distributed with a mean of 68g and standard deviation of 3.4g.
Each day 60 % of the muffins made are chocolate.
On a particular day, a muffin is randomly selected from all those made at the bakery.
(c)
(i) Find the probability that the randomly selected muffin weighs less than 61g.
(ii) Given that a randomly selected muffin weighs less than 61g, find the probability that it is chocolate.
The machine that makes the chocolate muffins is adjusted so that the mean weight of the chocolate muffins remains the same but their standard deviation changes to `sigma` g. The machine that makes the banana muffins is not adjusted. The probability that the weight of a randomly selected muffin from these machines is less than 61g is now 0.157.
(d) Find the value of `sigma`.
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Question 5
The Malthouse Charity Run is a 5 kilometre race. The time taken for each runner to complete the race was recorded. The data was found to be normally distributed with a mean time of 28 minutes and a standard deviation of 5 minutes.
A runner who completed the race is chosen at random.
(a) Write down the probability that the runner completed the race in more than 28 minutes.
(b) Calculate the probability that the runner completed the race in less than 26 minutes.
It is known that 20 % of the runners took more than 28 minutes and less than `k` minutes to complete the race.
(c) Find the value of `k`.
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Question 6
The price per kilogram of tomatoes, in euro, sold in various markets in a city is found to be normally distributed with a mean of 3.22 and a standard deviation of 0.84.
(a)
(i) On the following diagram, shade the region representing the probability that the price of a kilogram of tomatoes, chosen at random, will be higher than 3.22 euro.
(ii) Find the price that is two standard deviations above the mean price.
(b) Find the probability that the price of a kilogram of tomatoes, chosen at random, will be between 2.00 and 3.00 euro.
To stimulate reasonable pricing, the city offers a free permit to the sellers whose price of a kilogram of tomatoes is in the lowest 20%.
(c) Find the highest price that a seller can charge and still receive a free permit.
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Question 7
The marks achieved by students taking a college entrance test follow a normal distribution with mean 300 and standard deviation 100.
In this test, 10 % of the students achieved a mark greater than `k`.
(a) Find the value of `k`.
Marron College accepts only those students who achieve a mark of at least 450 on the test.
(b) Find the probability that a randomly chosen student will be accepted by Marron College.
(c) Given that Naomi attends Marron College, find the probability that she achieved a mark of at least 500 on the test.
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Question 8
The weights, in grams, of individual packets of coffee can be modelled by a normal distribution, with mean 102g and standard deviation 8g.
(a) Find the probability that a randomly selected packet has a weight less than 100g.
(b) The probability that a randomly selected packet has a weight greater than `w` grams is 0.444. Find the value of `w`.
(c) A packet is randomly selected. Given that the packet has a weight greater than 105g, find the probability that it has a weight greater than 110g.
(d) From a random sample of 500 packets, determine the number of packets that would be expected to have a weight lying within 1.5 standard deviations of the mean.
(e) Packets are delivered to supermarkets in batches of 80. Determine the probability that at least 20 packets from a randomly selected batch have a weight less than 95g.
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Question 9
It is known that 56% of Infiglow batteries have a life of less than 16 hours, and 94% have a life less than 17 hours. It can be assumed that battery life is modelled by the normal distribution `N(mu, sigma^2)`.
(a) Find the value of `mu` and the value of `sigma`.
(b) Find the probability that a randomly selected Infiglow battery will have a life of at least 15 hours.
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Question 10
In a large school, the heights of all fourteen-year-old students are measured.
The heights of the girls are normally distributed with mean 155cm and standard deviation 10cm.
The heights of the boys are normally distributed with mean 160cm and standard deviation 12cm.
(a) Find the probability that a girl is taller than 170cm.
(b) Given that 10% of the girls are shorter than `x` cm, find `x`.
(c) Given that 90% of the boys have heights between `q` cm and `r` cm where `q` and `r` are symmetrical about 160cm, and `q < r`, find the value of `q` and of `r`.
In the group of fourteen-year-old students, 60% are girls and 40% are boys.
The probability that a girl is taller than 170cm was found in part (a).
The probability that a boy is taller than 170cm is 0.202.
A fourteen-year-old student is selected at random.
(d) Calculate the probability that the student is taller than 170cm.
(e) Given that the student is taller than 170cm, what is the probability the student is a girl?
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Question 1
The flight times, `T` minutes, between two cities can be modelled by a normal distribution with a mean of 75 minutes and a standard deviation of `sigma` minutes.
(a) Given that 2% of the flight times are longer than 82 minutes, find the value of `sigma`.
(b) Find the probability that a randomly selected flight will have a flight time of more than 80 minutes.
(c) Given that a flight between the two cities takes longer than 80 minutes, find the probability that it takes less than 82 minutes.
On a particular day, there are 64 flights scheduled between these two cities.
(d) Find the expected number of flights that will have a flight time of more than 80 minutes.
(e) Find the probability that more than 6 of the flights on this particular day will have a flight time of more than 80 minutes.
(a) use of inverse normal to find the `z`-score
`z=2.0537...={82-75}/{sigma}`
`sigma=3.408401...=3.41`
(b) evidence of identifying the correct area under the normal curve
`P(T>80)=0.071193...=0.0712`
(c) recognition that `P(80 < T < 82)` is required
`P(T < 82| T > 80)= {P(80 < T < 82)}/{P(T > 80)}={0.051193...}/{0.071193...}=0.719075...`
`=0.719`
(d) recognition of binomial probability
`X ~ B(64, 0.071193...)`
`E(X) = 64 times 0.071193...=4.556353...`
`E(X)=4.56` (flights)
(e) `P(X > 6) = P(X >= 7)= 1- P(X <= 6)`
`=1-0.83088...`
`=0.1691196...`
`=0.169`
Question 2
The random variable `X` follows a normal distribution with mean `mu` and standard deviation `sigma`.
(a) Find `P(mu - 1.5 sigma < X < mu + 1.5 sigma)`.
The avocados grown on a farm have weights, in grams, that are normally distributed with mean `mu` and standard deviation `sigma`. Avocados are categorized as small, medium, large or premium, according to their weight. The following table shows the probability an avocado grown on the farm is classified as small, medium, large or premium.
The maximum weight of a small avocado is 106.2 grams.
The minimum weight of a premium avocado is 182.6 grams.
(b) Find the value of `mu` and of `sigma`.
A supermarket purchases all the avocados from the farm that weigh more than 106.2 grams.
(c) Find the probability that an avocado chosen at random from this purchase is categorized as
(i) medium;
(ii) large;
(iii) premium.
The selling prices of the different categories of avocado at this supermarket are shown in the following table:
The supermarket pays the farm $200 for the avocados and assumes it will then sell them in exactly the same proportion as purchased from the farm.
(d) According to this model, find the minimum number of avocados that must be sold so that the net profit for the supermarket is at least $438.
(a) `P( {mu -1.5 sigma - mu}/{sigma} < {X-mu}/{sigma} < {mu +1.5 sigma - mu}/{sigma})`
`P(-1.5 < Z < 1.5)` OR `1- 2 times P( Z < -1.5)`
`P(-1.5 < Z < 1.5) = 0.866385...`
`P( mu - 1.5 sigma < X < mu +1.5 sigma)=0.866`
(b) `z_1 =−1.75068...` and `z_2=1.30468...` (seen anywhere)
correct equation
`{106.2-mu}/{sigma}=−1.75068...,\ mu+1.30468...sigma=182.6`
attempt to solve their equations involving `z` values
`mu = 149.976..., sigma = 25.0051...`
`mu=150, sigma = 25.0`
(c)
(i) new sample space is 96% (may be seen in (ii) or (iii))
`P`(medium|not small) OR `0.576/0.96`
`P`(medium) `=0.6`
(ii) `P`(large) `=0.3`
(iii) `P`(premium) `=0.1`
(d) attempt to express revenue from avocados
`1.1 times 0.6 + 1.29 times 0.3 + 1.96 times 0.1` OR `1.243n`
correct inequality or equation for net profit in terms of `n`
`1.1 times 0.6n+1.29times 0.3n+1.96times0.1n−200 >= 438` OR `1.243n− 200 = 438`
attempt to solve the inequality
sketch OR `n = 513.274 ...`
`n=514`
Question 3
The time it takes Suzi to drive from home to work each morning is normally distributed with a mean of 35 minutes and a standard deviation of `sigma` minutes.
On 25 % of days, it takes Suzi longer than 40 minutes to drive to work.
(a) Find the value of `sigma`.
(b) On a randomly selected day, find the probability that Suzi’s drive to work will take longer than 45 minutes.
Suzi will be late to work if it takes her longer than 45 minutes to drive to work. The time it takes to drive to work each day is independent of any other day.
Suzi will work five days next week.
(c) Find the probability that she will be late to work at least one day next week.
(d) Given that Suzi will be late to work at least one day next week, find the probability that she will be late less than three times.
Suzi will work 22 days this month. She will receive a bonus if she is on time at least 20 of those days.
So far this month, she has worked 16 days and been on time 15 of those days.
(e) Find the probability that Suzi will receive a bonus.
(a) `T ~ N(35, sigma^2)`
`P(T > 40)=0.25` OR `P(T < 40)=0.75`
`z= 0.674489...`
valid equation using their score (clearly identified as `z`-score and not a probability)
`{40-35}/sigma=0.674489...` OR `5 = 0.674489 ... sigma`
`7.413011...`
`sigma = 7.41` min
(b) `P(T > 45)=0.0886718 = 0.0887`
(c) recognizing binomial probability
`L ~ B(5,0.0886718 ... )`
`P(L >= 1)=1-P(L=0)`
`0.371400 ...`
`P(L >= 1)=0.371`
(d) recognizing conditional probability in context
finding `{L < 3} cap {L >= 1} = {L = 1, L=2}` (may be seen in conditional probability)
`P(L = 1)+P(L=2)=0.36532...` (may be seen in conditional probability)
`P(L < 3| L >= 1) ={0.36532...}/{0.37140...}`
`0.983636 ...`
`0.984`
(e) recognizing that Suzi can be late no more than once (in the remaining six days)
`X ∼ B(6,0.0886718 ... )`, where `X` is the number of days late
`P(X <= 1) = P(X = 0) + P(X = 1) = 0.907294 ...`
`P`(Suzi gets a bonus) `= 0.907`
Question 4
A bakery makes two types of muffins: chocolate muffins and banana muffins.
The weights, `C` grams, of the chocolate muffins are normally distributed with a mean of 62g and standard deviation of 2.9g.
(a) Find the probability that a randomly selected chocolate muffin weighs less than 61g.
(b) In a random selection of 12 chocolate muffins, find the probability that exactly 5 weigh less than 61g.
The weights, `B` grams, of the banana muffins are normally distributed with a mean of 68g and standard deviation of 3.4g.
Each day 60 % of the muffins made are chocolate.
On a particular day, a muffin is randomly selected from all those made at the bakery.
(c)
(i) Find the probability that the randomly selected muffin weighs less than 61g.
(ii) Given that a randomly selected muffin weighs less than 61g, find the probability that it is chocolate.
The machine that makes the chocolate muffins is adjusted so that the mean weight of the chocolate muffins remains the same but their standard deviation changes to `sigma` g. The machine that makes the banana muffins is not adjusted. The probability that the weight of a randomly selected muffin from these machines is less than 61g is now 0.157.
(d) Find the value of `sigma`.
(a) `P(C < 61) = 0.365112 ... = 0.365`
(b) recognition of binomial e.g. `X ∼ B(12,0.365 ... )`
`P(X= 5) = 0.213666 ... = 0.214`
(c)
(i) Let CM represent ‘chocolate muffin’ and BM represent ‘banana muffin’
`P(B < 61) = 0.0197555 ...`
tree diagram showing two ways to have a muffin weigh `< 61`
`0.6 times 0.365 ... + 0.4 times 0.0197 ... = 0.226969 ... = 0.227`
(ii) recognizing conditional probability
`{0.6 times 0.365112 ...}/{0.226969 ...}=0.965183 ... = 0.965`
(d) `P(CM) times P(C < 61|CM) + P(BM) times P(B < 61|BM) = 0.157`
`(0.6 times P(C < 61) + (0.4 times 0.0197555 ... ) = 0.157`
`P(C < 61) = 0.248496 ...`
attempt to solve for `sigma` using GDC
`sigma = 1.47225 ...`
`sigma=1.47` (g)
Question 5
The Malthouse Charity Run is a 5 kilometre race. The time taken for each runner to complete the race was recorded. The data was found to be normally distributed with a mean time of 28 minutes and a standard deviation of 5 minutes.
A runner who completed the race is chosen at random.
(a) Write down the probability that the runner completed the race in more than 28 minutes.
(b) Calculate the probability that the runner completed the race in less than 26 minutes.
It is known that 20 % of the runners took more than 28 minutes and less than `k` minutes to complete the race.
(c) Find the value of `k`.
(a) `0.5 \ (1/2 ,50%)`
(b) `P(X <= 26)`
`0.345 \ (0.344578...,34.5%)`
(c) `0.7` OR `0.3` (seen)
`P(time < k) = 0.7` OR `P(time > k) = 0.3`
`(k =) 30.6 \ (30.6220 ... )` (minutes)
Question 6
The price per kilogram of tomatoes, in euro, sold in various markets in a city is found to be normally distributed with a mean of 3.22 and a standard deviation of 0.84.
(a)
(i) On the following diagram, shade the region representing the probability that the price of a kilogram of tomatoes, chosen at random, will be higher than 3.22 euro.
(ii) Find the price that is two standard deviations above the mean price.
(b) Find the probability that the price of a kilogram of tomatoes, chosen at random, will be between 2.00 and 3.00 euro.
To stimulate reasonable pricing, the city offers a free permit to the sellers whose price of a kilogram of tomatoes is in the lowest 20%.
(c) Find the highest price that a seller can charge and still receive a free permit.
(a)
(i)
(ii) 4.90
(b) 0.323 (0.323499...; 32.3 %)
(c) 2.51 (2.51303...)
Question 7
The marks achieved by students taking a college entrance test follow a normal distribution with mean 300 and standard deviation 100.
In this test, 10 % of the students achieved a mark greater than `k`.
(a) Find the value of `k`.
Marron College accepts only those students who achieve a mark of at least 450 on the test.
(b) Find the probability that a randomly chosen student will be accepted by Marron College.
(c) Given that Naomi attends Marron College, find the probability that she achieved a mark of at least 500 on the test.
(a)
428 (428.155...)
(b)
0.0668 (0.0668072..., 6.68%, 6.68072...%)
(c) `0.0228/0.0668 \ ({0.0227500...}/{0.0668072...})`
= 0.341 (0.340532..., 34.1%, 34.0532...%)
Question 8
The weights, in grams, of individual packets of coffee can be modelled by a normal distribution, with mean 102g and standard deviation 8g.
(a) Find the probability that a randomly selected packet has a weight less than 100g.
(b) The probability that a randomly selected packet has a weight greater than `w` grams is 0.444. Find the value of `w`.
(c) A packet is randomly selected. Given that the packet has a weight greater than 105g, find the probability that it has a weight greater than 110g.
(d) From a random sample of 500 packets, determine the number of packets that would be expected to have a weight lying within 1.5 standard deviations of the mean.
(e) Packets are delivered to supermarkets in batches of 80. Determine the probability that at least 20 packets from a randomly selected batch have a weight less than 95g.
(a) `X ∼ N(102,8^2)`
`P(X < 100) = 0.401`
(b) `P(X < w) = 0.444`
`=> w=103` (g)
(c) `P(X > 110|X > 105)={P(X > 110 cap X > 105)}/{P(X > 105)}={P(X > 110)}/{P(X > 105)}={0.15865...}/{0.35383...}`
`=0.448`
(d) `P(90 < X < 114) = 0.866 ...`
`0.866 ... times 500`
`=433`
(e) `p = P(X < 95) = 0.19078 ...`
recognizing `Y ∼ B(80, p)`
now using `Y ∼ B(80,0.19078 ... )`
`P(Y >= 20) = 0.116`
Question 9
It is known that 56% of Infiglow batteries have a life of less than 16 hours, and 94% have a life less than 17 hours. It can be assumed that battery life is modelled by the normal distribution `N(mu, sigma^2)`.
(a) Find the value of `mu` and the value of `sigma`.
(b) Find the probability that a randomly selected Infiglow battery will have a life of at least 15 hours.
(a) use of inverse normal (implied by ±0.1509... or ±1.554...)
`P(X < 16)=0.56 => {16-mu}/sigma=0.1509 ...`
`P(X < 17)=0.94 => {17-mu}/sigma=1.554 ...`
attempt to solve a pair of simultaneous equations
`\mu=15.9, sigma = 0.712`
(b) correctly shaded diagram or intent to find `P(X >= 15)`
`=0.895`
Question 10
In a large school, the heights of all fourteen-year-old students are measured.
The heights of the girls are normally distributed with mean 155cm and standard deviation 10cm.
The heights of the boys are normally distributed with mean 160cm and standard deviation 12cm.
(a) Find the probability that a girl is taller than 170cm.
(b) Given that 10% of the girls are shorter than `x` cm, find `x`.
(c) Given that 90% of the boys have heights between `q` cm and `r` cm where `q` and `r` are symmetrical about 160cm, and `q < r`, find the value of `q` and of `r`.
In the group of fourteen-year-old students, 60% are girls and 40% are boys.
The probability that a girl is taller than 170cm was found in part (a).
The probability that a boy is taller than 170cm is 0.202.
A fourteen-year-old student is selected at random.
(d) Calculate the probability that the student is taller than 170cm.
(e) Given that the student is taller than 170cm, what is the probability the student is a girl?
(a) `P(G > 170)=1−P(G < 170)`
`P(G < 170)=P(Z < {170-155}/10)`
`P(G > 170)= 1-Phi(1.5)=1 − 0.9332 = 0.0668`
(b) `z = −1.2816`
Correct calculation e.g. `x = 155 + (−1.282) times 10`
`x = 142`
(c) Calculating one variable
e.g. `P(B < r) = 0.95, z = 1.6449`
`r = 160 + 1.645(12) = 179.74 = 180`
Any valid calculation for the second variable, including use of symmetry
e.g. `P(B < q) = 0.05, z = −1.6449`
`r = 160 − 1.645(12) = 140.26 = 140`
(d) `P(M cap (B > 170)) = 0.4 × 0.2020,\ P(F cap (G > 170)) = 0.6 × 0.0668`
`P(H > 170)=0.0808+0.04008=0.12088=0.121` (3s.f.)
(e) `P(F|H > 170)={P(F cap (H > 170))}/{P(H > 170)}={0.60 times 0.0668}/0.121`
`(=0.0401/0.121 or 0.04008/0.1208)`
`=0.332`
Question 1
The flight times, `T` minutes, between two cities can be modelled by a normal distribution with a mean of 75 minutes and a standard deviation of `sigma` minutes.
(a) Given that 2% of the flight times are longer than 82 minutes, find the value of `sigma`.
(b) Find the probability that a randomly selected flight will have a flight time of more than 80 minutes.
(c) Given that a flight between the two cities takes longer than 80 minutes, find the probability that it takes less than 82 minutes.
On a particular day, there are 64 flights scheduled between these two cities.
(d) Find the expected number of flights that will have a flight time of more than 80 minutes.
(e) Find the probability that more than 6 of the flights on this particular day will have a flight time of more than 80 minutes.
Question 2
The random variable `X` follows a normal distribution with mean `mu` and standard deviation `sigma`.
(a) Find `P(mu - 1.5 sigma < X < mu + 1.5 sigma)`.
The avocados grown on a farm have weights, in grams, that are normally distributed with mean `mu` and standard deviation `sigma`. Avocados are categorized as small, medium, large or premium, according to their weight. The following table shows the probability an avocado grown on the farm is classified as small, medium, large or premium.
The maximum weight of a small avocado is 106.2 grams.
The minimum weight of a premium avocado is 182.6 grams.
(b) Find the value of `mu` and of `sigma`.
A supermarket purchases all the avocados from the farm that weigh more than 106.2 grams.
(c) Find the probability that an avocado chosen at random from this purchase is categorized as
(i) medium;
(ii) large;
(iii) premium.
The selling prices of the different categories of avocado at this supermarket are shown in the following table:
The supermarket pays the farm $200 for the avocados and assumes it will then sell them in exactly the same proportion as purchased from the farm.
(d) According to this model, find the minimum number of avocados that must be sold so that the net profit for the supermarket is at least $438.
Question 3
The time it takes Suzi to drive from home to work each morning is normally distributed with a mean of 35 minutes and a standard deviation of `sigma` minutes.
On 25 % of days, it takes Suzi longer than 40 minutes to drive to work.
(a) Find the value of `sigma`.
(b) On a randomly selected day, find the probability that Suzi’s drive to work will take longer than 45 minutes.
Suzi will be late to work if it takes her longer than 45 minutes to drive to work. The time it takes to drive to work each day is independent of any other day.
Suzi will work five days next week.
(c) Find the probability that she will be late to work at least one day next week.
(d) Given that Suzi will be late to work at least one day next week, find the probability that she will be late less than three times.
Suzi will work 22 days this month. She will receive a bonus if she is on time at least 20 of those days.
So far this month, she has worked 16 days and been on time 15 of those days.
(e) Find the probability that Suzi will receive a bonus.
Question 4
A bakery makes two types of muffins: chocolate muffins and banana muffins.
The weights, `C` grams, of the chocolate muffins are normally distributed with a mean of 62g and standard deviation of 2.9g.
(a) Find the probability that a randomly selected chocolate muffin weighs less than 61g.
(b) In a random selection of 12 chocolate muffins, find the probability that exactly 5 weigh less than 61g.
The weights, `B` grams, of the banana muffins are normally distributed with a mean of 68g and standard deviation of 3.4g.
Each day 60 % of the muffins made are chocolate.
On a particular day, a muffin is randomly selected from all those made at the bakery.
(c)
(i) Find the probability that the randomly selected muffin weighs less than 61g.
(ii) Given that a randomly selected muffin weighs less than 61g, find the probability that it is chocolate.
The machine that makes the chocolate muffins is adjusted so that the mean weight of the chocolate muffins remains the same but their standard deviation changes to `sigma` g. The machine that makes the banana muffins is not adjusted. The probability that the weight of a randomly selected muffin from these machines is less than 61g is now 0.157.
(d) Find the value of `sigma`.
Question 5
The Malthouse Charity Run is a 5 kilometre race. The time taken for each runner to complete the race was recorded. The data was found to be normally distributed with a mean time of 28 minutes and a standard deviation of 5 minutes.
A runner who completed the race is chosen at random.
(a) Write down the probability that the runner completed the race in more than 28 minutes.
(b) Calculate the probability that the runner completed the race in less than 26 minutes.
It is known that 20 % of the runners took more than 28 minutes and less than `k` minutes to complete the race.
(c) Find the value of `k`.
Question 6
The price per kilogram of tomatoes, in euro, sold in various markets in a city is found to be normally distributed with a mean of 3.22 and a standard deviation of 0.84.
(a)
(i) On the following diagram, shade the region representing the probability that the price of a kilogram of tomatoes, chosen at random, will be higher than 3.22 euro.
(ii) Find the price that is two standard deviations above the mean price.
(b) Find the probability that the price of a kilogram of tomatoes, chosen at random, will be between 2.00 and 3.00 euro.
To stimulate reasonable pricing, the city offers a free permit to the sellers whose price of a kilogram of tomatoes is in the lowest 20%.
(c) Find the highest price that a seller can charge and still receive a free permit.
Question 7
The marks achieved by students taking a college entrance test follow a normal distribution with mean 300 and standard deviation 100.
In this test, 10 % of the students achieved a mark greater than `k`.
(a) Find the value of `k`.
Marron College accepts only those students who achieve a mark of at least 450 on the test.
(b) Find the probability that a randomly chosen student will be accepted by Marron College.
(c) Given that Naomi attends Marron College, find the probability that she achieved a mark of at least 500 on the test.
Question 8
The weights, in grams, of individual packets of coffee can be modelled by a normal distribution, with mean 102g and standard deviation 8g.
(a) Find the probability that a randomly selected packet has a weight less than 100g.
(b) The probability that a randomly selected packet has a weight greater than `w` grams is 0.444. Find the value of `w`.
(c) A packet is randomly selected. Given that the packet has a weight greater than 105g, find the probability that it has a weight greater than 110g.
(d) From a random sample of 500 packets, determine the number of packets that would be expected to have a weight lying within 1.5 standard deviations of the mean.
(e) Packets are delivered to supermarkets in batches of 80. Determine the probability that at least 20 packets from a randomly selected batch have a weight less than 95g.
Question 9
It is known that 56% of Infiglow batteries have a life of less than 16 hours, and 94% have a life less than 17 hours. It can be assumed that battery life is modelled by the normal distribution `N(mu, sigma^2)`.
(a) Find the value of `mu` and the value of `sigma`.
(b) Find the probability that a randomly selected Infiglow battery will have a life of at least 15 hours.
Question 10
In a large school, the heights of all fourteen-year-old students are measured.
The heights of the girls are normally distributed with mean 155cm and standard deviation 10cm.
The heights of the boys are normally distributed with mean 160cm and standard deviation 12cm.
(a) Find the probability that a girl is taller than 170cm.
(b) Given that 10% of the girls are shorter than `x` cm, find `x`.
(c) Given that 90% of the boys have heights between `q` cm and `r` cm where `q` and `r` are symmetrical about 160cm, and `q < r`, find the value of `q` and of `r`.
In the group of fourteen-year-old students, 60% are girls and 40% are boys.
The probability that a girl is taller than 170cm was found in part (a).
The probability that a boy is taller than 170cm is 0.202.
A fourteen-year-old student is selected at random.
(d) Calculate the probability that the student is taller than 170cm.
(e) Given that the student is taller than 170cm, what is the probability the student is a girl?
