At a certain university, 15\% of students study psychology. 2000 random students have been asked what subject they are studying. Of those asked, 12\% were psychology students.
State the population.
Calculate the value of the population proportion.
Calculate the value of the sample proportion.
Do the 2000 students tested represent a simple random sample?
A survey was carried out to investigate the number of teachers in Australian schools who like using the chalkboard to teach. This survey found that in a sample of 1222 teachers, 321 liked using the chalkboard, while the rest did not.
Calculate the sample proportion of \hat{p}, of those teachers surveyed who like using the chalkboard. Round your answer to three decimal places.
Hence, estimate the standard deviation, \hat{\sigma}, of the random variable \hat{P}, for such samples of size 1222. Round your answer to three decimal places.
Lachlan wanted to know the proportion of commuters that regularly listen to music on his train. In his carriage, he found that in a sample of 35 commuters, 7 were listening to music.
Calculate the sample proportion, \hat{p}, of people who were listening to music in his carriage.
Hence, estimate the standard deviation, \hat{\sigma}, of the random variable \hat{P}, for such samples of size 35. Round your answer to three decimal places.
A survey involving 218 midwives found that 150 of them were aged between 46 and 60 years.
Calculate the sample proportion, \hat{p}, of those midwives surveyed who were aged between 46 and 60 years.
Hence, estimate the standard deviation, \hat{\sigma}, of the random variable \hat{P}, for such samples of size 218. Round your answer to three decimal places.
To practice her archery, Sandy marked a small interval on the board with length 5 \text{ cm}. Sandy takes 240 shots at the board from a certain distance, with 151 landing between the small interval.
Calculate the sample proportion, \hat{p}, of shots that landed in the 5 \text{ cm} interval.
Hence, estimate the standard deviation, \hat{\sigma}, of the random variable \hat{P}, for such samples of size 240. Round your answer to three decimal places.
The proportion of the population exhibiting a certain characteristic is p = 0.69. A sample size of 50 was taken from the population.
Determine the standard deviation, \sigma_{\hat{P}}, of the random variable \hat{P}, which measures the sample proportion of those exhibiting a certain characteristic. Round your answer to three decimal places.
Using a normal approximation, what is the probability that \hat{P} would lie between 0.64 and 0.78? Round your answer to two decimal places.
The proportion of individuals in a small, country town who own a car is p = 0.16. A sample size of 30 were surveyed from the population.
Determine the standard deviation, \sigma_{\hat{P}}, of the random variable \hat{P}, which measures the proportion of those that own a car in the sample. Round your answer to three decimal places.
Using a normal approximation, what is the probability that \hat{P} would lie between 0.11 and 0.2? Round your answer to two decimal places.
The proportion of voters in the population who favour Candidate A is 50\%. Out of a random sample of 487 voters, 248 indicated that they would vote for Candidate A at the next election.
What is the value of the proportion, \hat{p}, of those that would vote for Candidate A in the random sample? Round your answer to three decimal places.
Using the normal approximation, find the probability that, in any random sample of size 487 voters, the proportion who favour Candidate A is greater than the value of \hat{p} observed in this particular sample. Round your answer to two decimal places.
The probability of an adult having fair hair is p = 0.2. Many samples of size 381 were taken from the population.
Estimate the mean of this empirical distribution.
Would the distribution of the sample proportions be approximately normal? Explain your answer.
State the standard deviation, \sigma, of \hat{P}, which is equal to the proportions of adults with fair hair in any sample of size 381. Round your answer to three decimal places.
Using the normal approximation, determine the probability, q, that in any random sample of 381 adults, there are between 70 and 84 people with fair hair. Round your answer to two decimal places.
4\% of Melbourne students travel to school by tram. Many random samples of students with size 181 were surveyed.
Estimate the mean of this empirical distribution.
State the standard deviation, \sigma, of \hat{P}, which is equal to the proportion of Melbourne students who travel to school by tram in any sample of size 181. Round your answer to three decimal places.
Using the normal approximation, determine the probability, q, that in any random sample of 181 Melbourne students, there are between 4 and 11 people who travel to school via the tram. Round your answer to two decimal places.
A machine has probability of p = 0.1 of producing a defective item. The next batch of 1000 was sampled.
Determine the standard deviation, \sigma_{\hat{P}}, of the random variable \hat{P}, which measures the proportion of defective items in the batch. Round your answer to three decimal places.
Using a normal approximation, what is the probability that \hat{P} would lie between 0.09 and 0.11? Round your answer to two decimal places.
Using a normal approximation, what is the probability that in the batch of 1000, the proportion of defective items, \hat{P}, will be between 0.09 and 0.11, given that you know it is greater than 0.1? Round your answer to two decimal places.
In a population it is known that p\% prefer a certain colour. In a sample of size n from this population it is known that \hat{p} prefer this same colour. Assume the sample size n is large.
Describe what should be done if we wish to determine the probability that in a future sample of size n, the proportion of the sample who prefer this colour is less than 0.3.
In a particular city, it is known that 43\% of residents use public transport at least once a week. A random sample of 300 people from this city finds that 122 use public transport at least once a week.
Determine the value of the sample proportion.
Describe what should be done if we wish to determine the probability of future samples of size 300 having a proportion greater than 0.45 who use public transport at least once a week.
Hence, determine the approximate probability that in a random sample of 300 residents from this city, the proportion who used public transport at least once a week is greater than 0.45. Round your answer to two decimal places.
From a population of people who shop at a certain shopping centre, it is known that 58\% shop at the grocer. A random sample of 500 people at this shopping centre is taken and 270 were found to have shopped at the grocer.
Determine the value of the sample proportion.
If we wish to determine the probability that a future sample of size 500 has a proportion greater than \dfrac{270}{500}, find the following for the normal distribution that can be used to best approximate this, correct to two decimal places:
Mean
Standard deviation
Hence, determine the approximate probability that in a random sample of 500 shoppers, the proportion who shop at the grocer is greater than \dfrac{270}{500}. Round your answer to two decimal places.
The proportion of ATAR students studying Mathematics Methods in WA is 28\%. A particular Western Australian school, Swan High, has 48 students studying Mathematics Methods out of their 150 ATAR students.
Find the following for the normal distribution that can be used to best approximate this, rounding your answers to two decimal places:
Mean
Standard deviation
Hence, determine the approximate probability that in 180 randomly chosen ATAR students, the proportion studying Mathematics Methods is less that than of Swan High. Round your answer to three decimal places.
It is known that 22\% of households in a large city own a dog. Let X be the random variable that represents the number of dog owning households chosen in a random sample of 120 households.
If we model X using a binomial distribution, state the parameter values of the following:
n
p
Using the binomial model, find the probability that less than 30 dog owning households will be chosen in the random sample of 120 households. Round your answer to three decimal places.
If we were to take a large number of samples of 120 households and calculate the sample proportion for each, estimate the mean of this empirical distribution of sample proportions.
Find the following for the normal distribution that best approximates the distribution of the sample proportions, rounding your answers to two decimal places:
Mean
Standard deviation
83\% of Australian television viewers regularly watch the news. Let X be the random variable that represents the number of people who watch the news in a random sample of 50 Australian television viewers.
If we model X using a binomial distribution, state the parameter values of the following:
n
p
Using the binomial model, calculate the probability that at least 40 people watch the news in the random sample of 50 Australian television viewers. Round your answer to three decimal places.
If we were to take a large number of samples of 50 Australian television viewers and calculate the sample proportion for each, estimate the mean of this empirical distribution of sample proportions.
Find the following for the normal distribution that best approximates the distribution of the sample proportions, rounding your answers to two decimal places:
Mean
Standard deviation
The waiting time at a particular bus stop is uniformly distributed between 0 and 7 minutes.
Find the probability that the waiting time for a particular person at this bus stop is no more than 3 minutes.
A random sample of 70 people using this bus stop is taken, and the number of people X, waiting no more than 3 minutes is taken. If we model X using a binomial distribution, state the parameter values of the following:
n
p
If we were to take a large number of samples of 70 individuals waiting at the bus stop and calculate the sample proportion that waited no more than 3 minutes for each, estimate the mean of this empirical distribution of sample proportions.
Find the following for the normal distribution that best approximates the distribution of the sample proportions, rounding answers to two decimal places:
Mean
Standard deviation
The time taken to download a movie is uniformly distributed between 10 and 15 minutes.
Find the probability that the download time for a particular movie is less than 13 minutes.
Many random samples, each containing 40 movie downloads are taken and each sample is used to calculate an estimate for the proportion of movies that took less than 13 minutes to download.
Estimate the mean of this empirical distribution of the sample proportions of downloads that took less than 13 minutes.
Find the following for the normal distribution that best approximates the distribution of the sample proportions, rounding your answers to two decimal places:
Mean
Standard deviation
Hence, calculate the probability that a randomly chosen sample has a sample proportion of movies taking less than 13 minutes to download greater than 0.5.
The mass of flour in a 1 \text{-kg} pack is normally distributed with a mean of 997 \text{ g} and a standard deviation of 4 \text{ g}.
Find the probability that the mass of flour in a randomly chosen pack is greater than stated, correct to three decimal places.
A random sample of 50 \, 1\text{-kg} flour packs is taken, and the number of packs X, weighing more than stated is recorded. If we model X using a binomial distribution, state the parameter values of the following, rounded to three decimal places:
n
p
If we were to take a large number of samples of 50 flour packs and calculate the sample proportion that weighed more than stated for each, estimate the mean of this empirical distribution of the sample proportions.
Find the following for the normal distribution that best approximates the distribution of the sample proportions, rounding your answers to three decimal places:
Mean
Standard deviation
A proportion p of high school students in Melbourne catch a bus to school. The standard deviation of sample proportions of students who take the bus to the school in a sample of size 150 is \dfrac{\sqrt{2}}{40}. Find the possible values of p.
A proportion p of Tasmanians live in rural areas. The standard deviation of sample proportions of Tasmanians who live in rural areas in a sample of size 200 is \dfrac{1}{30}. Find the possible values of p.
The heights of people in Milan is normally distributed with a mean of 177 \text{ cm} and a standard deviation of 5 \text{ cm}.
Find the probability that the height of a randomly chosen person is less than 170 \text{ cm}. Round your answer to two decimal places.
Consider the proportion of a random sample of 50 people that are less than 170 \text{ cm}. If this experiment is repeated many times, estimate the mean of this empirical distribution of the sample proportions.
Find the following for the normal distribution that best approximates the distribution of the sample proportions, rounding your answers to two decimal places:
Mean
Standard deviation
Hence, calculate the probability that a randomly chosen sample of size 50 has a sample proportion of people less than 170 \text{ cm} that is at least 0.04.
Determine the minimum size of the sample n if the standard deviation of the sampling distribution of the proportion of people less than 170 \text{ cm} is not to exceed 0.03.
It is known that the proportion of Australians with extras-only private health insurance is 9.2\%. A poll is taken to estimate the proportion of Australians buying extras-only private health insurance, and the size of the sample is 2500.
Find the following for the normal distribution that best approximates the distribution of the sample proportion of Australians with extras-only private, rounding your answers to three decimal places:
Mean
Standard deviation
Estimate the probability that the sample proportion is less than 0.09, correct to two decimal places.
Estimate the probability that between 9\% and 10\% of people in the sample have extras-only private health insurance. Round your answer to two decimal places.
Estimate the probability that the proportion of extras-only private health insurance holders in the sample differs by more than 0.01 from 0.092.