From a sample of 400 young workers in Seattle, 37 had amber eyes. What is the sample proportion of amber eyes for young workers in Seattle?
A manufacturer of cement trucks checked some of the cement trucks being distributed for faults. On one day, 75 cement trucks were checked and 16 of these were found to have some kind of fault. What was the sample proportion for faulty cement trucks?
At an electronics store, cameras are sold at prices ranging from \$150 to \$630. On a particular day, there are 106 cameras on sale and 25 of these cost less than \$260. If Irene came into the store and selected a camera at random, estimate the probability that camera would cost less than \$260.
A survey of 115 randomly selected people in Busan found that 6 of them were aged over 55. A second survey of 2183 randomly selected people in Busan found that 475 of them were aged over 55.
What is the sample proportion of people in Busan over the age of 55 for the first survey?
What is the sample proportion of people in Busan over the age of 55 for the second survey?
Which sample proportion is likely to be the better estimate of the population proportion? Explain your answer.
Phinosis Pty. Ltd. monitors the population of chameleons in the local area twice every month. Last year, a number of the chameleons were tagged with brown tags.
The first survey in March involves 551 chameleons and 110 of these have brown tags. In this survey, what is the sample proportion of chameleons with brown tags?
The second survey in March involves 203 chameleons and 15 of these have brown tags. In this survey, what is the sample proportion of chameleons with brown tags?
Assume that there is no change in the population of chameleons between the two surveys. Which sample proportion is likely to be the better estimate of the population proportion? Explain your answer.
A population is sampled several times to investigate the proportion that purchase a new pot plant each year. Each sample is equal in size and the values of the sample proportions are recorded in the given graph:
How many samples were taken?
Hence, estimate the population proportion of customers who purchased a new pot plant, correct to three decimal places.
Several samples of a population are surveyed. Each sample is equal in size and the values of the sample proportions are recorded in the following graph:
How many samples were taken?
Hence, estimate the population proportion, correct to three decimal places.
A census for a particular country showed that 94\% of people used public transport at some point during a regular week. At about the same time as the census, a sample of 2420 people in a region of the country showed that 1001 of those people used public transport at some point during a regular week.
Determine p, the population proportion of the residents who use public transport at least once a week.
Determine \hat{p}, the sample proportion of the residents who use public transport at least once a week. Round your answer to two decimal places.
Comparing the population and sample proportions, state whether the sample exhibits bias and is indicative of the larger population.
A census for a particular country showed that 28\% of people used public transport at some point during a regular week. At about the same time as the census, a sample of 5500 people in a region of the country showed that 1573 of those people used public transport at some point during a regular week.
Determine p, the population proportion of the residents who use public transport at least once a week.
Determine \hat{p}, the sample proportion of the residents who use public transport at least once a week. Round your answer to two decimal places.
Comparing the population and sample proportions, state whether the sample exhibits bias and is indicative of the larger population.
The spinner below was spun 160 times and the number 2 was spun 56 times:
Determine p, the population proportion of landing on 2.
Determine \hat{p}, the sample proportion of landing on 2. Round your answer to two decimal places.
What could be done to to increase the accuracy of the estimated population proportion?
A dog has three puppies. Let M represent the number of male puppies in this litter.
If a dog has 3 puppies, then the number of male puppies, M, can be 0, 1, 2 or 3. Determine the values of the proportions, \hat{P} of male puppies in the litter associated with each outcome of M:
M = 0
M = 1
M = 2
M = 3
Complete the probability distribution table for M and \hat{P}:
| m | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| P\left(M=m\right) | \dfrac{1}{8} | |||
| \hat{p} | 0 | \dfrac{1}{3} | \dfrac{2}{3} | 1 |
| P(\hat{P}=\hat{p}) | \dfrac{3}{8} |
Find P\left(\hat{P} \gt \dfrac{1}{2}\right).
Three marbles are randomly drawn from a bag containing five black and six grey marbles. Let X be the number of black marbles drawn, with replacement.
What is p, the proportion of black marbles in the bag?
If 3 marbles are drawn, with replacement, then the number of black marbles drawn, X, can be 0, 1, 2 or 3. Determine the values of the sample proportions, \hat{P}, of black marbles associated with each outcome of X:
X = 0
X = 1
X = 2
X = 3
Complete the probability distribution table for X and \hat{P}, rounding each probability to four decimal places:
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| P\left(X=x\right) | 0.1623 | |||
| \hat{p} | 0 | \dfrac{1}{3} | \dfrac{2}{3} | 1 |
| P\left(\hat{P}=\hat{p}\right) | 0.4057 |
Find P\left(\hat{P} \lt \dfrac{1}{2}\right), correct to four decimal places.
A company wants to know the likelihood of securing sales with potential clients. Historically, the company has 70 successful sales for every 100 potential clients contacted. Let X be the number of sales the company secures within the next 4 potential clients.
What is p, the proportion of successful sales?
If 4 potential clients are called, then the number of successful sales, X, can be 0, 1, 2, 3 or 4. Determine the values of the sample proportions, \hat{P} of successful sales associated with each outcome of X:
X = 0
X = 1
X = 2
X = 3
X = 4
Complete the probability distribution table for X and \hat{P}, rounding each probability to four decimal places:
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| P\left(X=x\right) | 0.0081 | ||||
| \hat{p} | 0 | 0.25 | 0.5 | 0.75 | 1 |
| P\left(\hat{P}=\hat{p}\right) | 0.0756 |
Find P\left(\hat{P} \gt 0.25\right) to four decimal places.
A pencil case contains 9 red pens and 7 black pens. 4 pens are drawn randomly from the pencil case, one at a time, each being replaced before the next one is drawn. Let W be the number of red pens drawn.
What is p, the proportion of red pens in the pencil case?
If 4 pens are drawn, then the number of red pens drawn, W, can be 0, 1, 2, 3 or 4. Determine the values of the sample proportions, \hat{P} of red pens, associated with each outcome of W:
W = 0
W = 1
W = 2
W = 3
W = 4
Complete the probability distribution table for W and \hat{P}, rounding each probability to four decimal places:
| w | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| P\left(W=w\right) | 0.0366 | ||||
| \hat{p} | 0 | 0.25 | 0.5 | 0.75 | 1 |
| P\left(\hat{P}=\hat{p}\right) | 0.1884 |
Find P \left( \hat{P} < 0.5 \right), correct to four decimal places.
A pencil case contains 6 red pens and 5 black pens. 4 pens are drawn without replacement. Let X be the number of black pens.
What is p, the proportion of black pens in the pencil case?
Complete the probability distribution table for X and \hat{P}, rounding each probability to two decimal places:
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| P\left(X=x\right) | 0.05 | ||||
| \hat{p} | |||||
| P\left(\hat{P}=\hat{p}\right) | 0.30 |
Find P \left( \hat{P} < \dfrac{1}{2} \right), correct to two decimal places.
In Victoria, it has been shown that 40\% of all voters are in favour of daylight saving. A sample of 5 voters are selected from Victoria at random.
Write down the possible values of the sample proportion, \hat{P}, of individuals that are in favour of daylight saving in the sample.
Construct a probability distribution table which summarises the sample proportion of individuals from Victoria who favoured daylight saving. Round the probabilities to four decimal places.
Find P\left(\hat{P} \lt \dfrac{3}{5}\right), correct to four decimal places.
In a primary school, it is known that 27\% of students have allergies and will always show serious allergic reactions during the start of Spring.
Four students enter the school on the first day of Spring. Write down the possible values of the sample proportion, \hat{P}, of students who have an allergic reaction.
Construct a probability distribution table which summarises the sample proportion of students who may have an allergic reaction. Round the probabilities correct to four decimal places.
Find P\left(\hat{P}\lt 0.6 \right), correct to four decimal places.
A Facebook advertising campaign gets an average of 200 likes per 1000 users who see the advertisement.
What is p, the proportion of users who like the advertisement?
If 5 people targeted by the campaign were selected and questioned, write down the possible values of the sample proportion, \hat{P}, of people who clicked “Like”.
Construct a probability distribution table which summarises the sample proportion of people who clicked “Like” in this sample.
Find P\left(\hat{P} \lt \dfrac{1}{2} \, \vert \, \hat{P} \geq 0 \right).
In the past, when a gym ran promotions offering a one month free membership, they found that 6 out of 10 sign-ups became full time members of the gym.
What is p, the proportion of individuals who became full time members after signing up for the promotion?
Five people who recently signed up for the promotion were sampled randomly at the gym. Write down the possible values of the sample proportion, \hat{P}, of people who will become full time members.
Construct a probability distribution table which summarises the sample proportion of people who will become full time members in the sample.
Find P\left(\hat{P} \lt \dfrac{1}{3} \vert \hat{P} \geq 0 \right).
The founders of a dating app claim that 38 out of 100 messages are replied to.
What is p, the proportion of messages that get replies? Write your answer in simplified fraction form.
Assuming that this proportion is true, if Edward sends 5 messages to different people, then write down the possible values of the sample proportion, \hat{P}, of people who reply.
Construct a probability distribution table which summarises the sample proportion of people who replied to Edward. Round the probabilities to four decimal places.
Find P\left(\hat{P} \gt \dfrac{3}{4} \vert \hat{P} \geq 0 \right), correct to four decimal places.
Two dice are rolled and the absolute value of the differences between the numbers appearing uppermost are recorded.
Complete the table below that represents the sample space:
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | 0 | 3 | ||||
| 2 | 1 | |||||
| 3 | 2 | |||||
| 4 | ||||||
| 5 | 4 | 2 | ||||
| 6 |
Let X be defined as the absolute value of the difference between the two dice. Construct the probability distribution for X.
What is the probability, p, that X \gt 3?
Two dice were rolled 3 times. Their absolute difference was recorded. Let Y be the number of times the absolute difference was greater than 3. Then Y can be 0, 1, 2 or 3.
Determine \hat{P}, the sample proportion of absolute differences greater than 3 associated with each of the following outcome of Y:
Y = 0
Y = 1
Y = 2
Y = 3
Complete the probability distribution for Y and \hat{P} below, rounding each probability to four decimal places:
| y | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| P\left(Y=y\right) | 0.0694 | |||
| \hat{p} | ||||
| P\left(\hat{P}=\hat{p}\right) | 0.0046 |
Find P\left(\hat{P} \lt 1\right), correct to four decimal places.
Over the course of one day, ten samples of customers at a cafe were surveyed. Each sample involved 25 customers who had ordered at least one item from the menu.
The number of customers in each sample who ordered a tea is shown in the table below. Complete the table by calculating the sample proportion for each sample:
| Sample | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| \text{Ordered a tea} | 18 | 17 | 24 | 16 | 11 | 9 | 14 | 13 | 3 | 9 |
| \text{Proportion} |
Calculate the mean of the sample proportions, correct to three decimal places.
Hence, estimate the population proportion of customers who ordered a tea, correct to three decimal places.
At an awards ceremony for Sun Valley Grammar School, Neville wants to find out how many students studied Geography. To do this, he surveys several samples of 40 students.
The number of students in each sample who studied Geography is shown in the table below. Complete the table by calculating the sample proportion for each sample:
| Sample | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| \text{Geography} \\ \text{students} | 15 | 32 | 9 | 15 | 25 | 33 | 15 | 22 | 26 | 23 |
| \text{Proportion} |
Calculate the mean of the sample proportions, correct to three decimal places.
Hence, estimate the population proportion of students who studied Geography, correct to three decimal places.
In Italy, 37\% of Italians also speak English. If a sample of 200 Italians in Italy are selected at random, determine:
The expected number in the sample who speak English.
The expected sample proportion of English speakers.
The standard deviation of the sample proportions of English speakers, correct to two decimal places.
15\% of all customers at a book store bought at least one autobiography. In a random sample of 100 customers, determine:
The expected number of customers who purchased an autobiography.
The expected sample proportion of customers who purchased an autobiography.
The standard deviation of the sample proportions of customers who purchased an autobiography, correct to two decimal places.
On any day of the year, approximately 42\,000 of the 400\,000 people residing in the city of Florence are tourists. Determine, in a random sample of 600 people:
The expected number of tourists.
The expected sample proportion of tourists.
The standard deviation of the sample proportions of tourists to three decimal places.
A social media advertising campaign gets an average of 150 follows per 1000 users who see the advertisement. Let X be the number of follows and \hat{P} be the proportion of follows in 25 different views of the advertisement.
Determine E \left(X\right).
Determine the standard deviation of X.
Determine E \left( \hat{P} \right).
Determine the standard deviation of \hat{P}.
Three marbles are drawn with replacement from a bag containing six black and five grey marbles. Let X be the number of black marbles drawn and \hat{P} be the proportion of black marbles drawn.
Determine E \left(X\right).
Determine the standard deviation of X.
Determine E \left( \hat{P} \right).
Determine the standard deviation of \hat{P}.
Laura always has 5 pairs of gloves in her drawer. Each glove isn't necessarily placed with its pair. Each morning she randomly chooses two gloves, one at a time, from the drawer in the dark to take to work. Let X be the number of left-hand gloves chosen, without replacement.
Determine the values of the sample proportion, \hat{P}, of left-hand gloves associated with each value of X:
x = 0
x = 1
x = 2
Complete the probability distribution table for X and \hat{P}, rounding each probability to two decimal places:
Calculate E \left( \hat{P} \right).
Calculate \text{Var} \left( \hat{P} \right).
| x | 0 | 1 | 2 |
|---|---|---|---|
| P\left(X=x\right) | |||
| \hat{p} | |||
| P\left(\hat{P}=\hat{p}\right) |
In group A, there are 150 people leaving from Australia to fly overseas in January and 12\% are flying to the USA. In group B, there are 150 people leaving from Australia to fly overseas in January and 21 are flying to the USA.
In group A, how many people are flying to the USA?
What is the proportion of group B flying to the USA?
Would the mean of the sample proportions drawn from group A be larger, smaller or the same as the mean from group B?
If we compare the standard deviation of sample proportions drawn from each group, with equal sample size, would the standard deviation from group A be larger, smaller or the same as the standard deviation from group B?
In group A, there are 200 people questioned as to whether they use gas heating in winter and 45\% answered "Yes". In group B, a further 200 people are asked the same question and 84 answered "Yes".
In group A, how many people in this sample answered "Yes"?
What is the proportion of group B that answered "Yes"?
Would the mean of the sample proportions drawn from group A that said "Yes" be larger, smaller or the same as the mean from group B that said "Yes"?
If we compare the standard deviation of sample proportions drawn from each group that said "Yes", with equal sample size, would the standard deviation from group A be larger, smaller or the same as the standard deviation from group B?
It is known that 25\% of Australian drivers prefer to drive a manual car. If a random sample of 80 Australian drivers are chosen at random, determine:
The probability that the sample proportion is equal to the population proportion, correct to three decimal places.
The expected sample proportion of drivers preferring a manual car.
The standard deviation of the sample proportions of drivers preferring a manual car, correct to three decimal places.
The probability that the sample proportion lies within one standard deviation of the population proportion, correct to two decimal places.
The probability that the sample proportion lies within two standard deviations of the population proportion, correct to two decimal places.
It is known that of the 1400 students enrolled in education at UWA, 70 are studying to become mathematics teachers. If a random sample of 60 UWA education students are chosen at random, determine:
The probability that the sample proportion is equal to the population proportion, correct to three decimal places.
The expected sample proportion of education students becoming maths teachers.
The standard deviation of the sample proportions of education students becoming maths teachers, correct to three decimal places.
The probability that the sample proportion lies within one standard deviation of the population proportion correct to three decimal places.
The probability that the sample proportion lies within two standard deviations of the population proportion, correct to three decimal places.