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 UQ, 70 are studying to become mathematics teachers. If a random sample of 60 UQ 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.