How many QCAA Mathematical Methods questions cover Sampling and proportions?

AusGrader has 39 QCAA Mathematical Methods questions on Sampling and proportions, all with instant AI grading and detailed marking feedback.

QCAA (QCE) Mathematical Methods — Sampling and proportions Questions & Answers | AusGrader

Q7

2023

QCAA

Paper 2

1 mark

Q7

1 mark

The distribution of a certain sample proportion has a mean of 0.70 and a standard deviation of 0.02.
Determine the sample size.

A

525

B

750

C

1750

D

2500

Q10

2022

QCAA

Paper 1

1 mark

Q10

1 mark

A survey plans to draw conclusions based on a random sample of 1% of Queensland's adult population.
To be regarded as a random sample, every

A

adult in the population will be placed in an alphabetical list and every 100th person will be selected for the sample.

B

adult in the population can choose to participate until the sample size has been reached.

C

subgroup within the population will be represented in a similar proportion in the sample.

D

adult in the population will have an equal chance of being selected for the sample.

Q16

2024

QCAA

Paper 2

4 marks

Q16

4 marks

At council meetings in a particular town, new proposals are only discussed if more than 80% of the community are in favour of the proposal.
To discover community opinion on a new bus route proposal, the council conducted several surveys, each with a sample size of 120. The distribution of the sample proportions from the surveys had a standard deviation of 0.04.
Make a justified decision as to whether the new bus route proposal would be discussed at a council meeting.

Analytical procedure:

$\sigma=\sqrt{\dfrac{p(1-p)}{n}}$ (formula book)

$0.04=\sqrt{\dfrac{p(1-p)}{120}}$

$0.0016=\dfrac{p(1-p)}{120}$

$0.192=p(1-p)$

$0.192=p-p^2$

$p^2-p+0.192=0$

Use GDC equation solver or graph $y=0.04$ and the
square root function to find intersections:

$p=0.25921$

$=26\%$

or

$p=0.7408$

$=74\%$

Both population proportions (26% and 74%) are less than
80%.

Therefore, the bus route proposal would not be discussed
at a council meeting.

Marking Criteria

Descriptor	Marks
Correctly substitutes the given information into the standard deviation formula for sample proportion	1
Determines a possible value for the population proportion	1
Determines a second possible population proportion value	1
Makes a justified decision regarding the proposal	1

Q14

2022

QCAA

Paper 2

8 marks

Q14

Ravi randomly sampled 200 different pet owners in Brisbane and found that 50 celebrate their pet's birthday.

Q14a

2 marks

Determine an approximate 95% confidence interval for the proportion of Brisbane pet owners who celebrate their pet's birthday.

Using GDC to determine confidence interval associated with $n = 200, \hat{p} = 0.25, z = 1.96$

$(0.19, 0.31)$

Marking Criteria

Descriptor	Marks
correctly identifies all of the information required to establish the confidence interval	1
correctly determines the confidence interval	1

Q14b

2 marks

Two of Ravi's friends also randomly sampled Brisbane pet owners. The results are shown in the table.

Friend's name	Number sampled	Number who celebrate their pet's birthday
Khadija	100	26
Tim	150	34

Khadija suggested a more precise estimate for the proportion of Brisbane pet owners who celebrate their pet's birthday could be obtained by combining their results.

Using all available data, determine an approximate 95% confidence interval for the proportion of Brisbane pet owners who celebrate their pet's birthday.

Combining results
$n = 450, \hat{p} = \frac{11}{45}$

Using GDC
$(0.2047, 0.2842)$

Marking Criteria

Descriptor	Marks
correctly determines $n$ and $\hat{p}$ for the combined sample	1
determines confidence interval	1

Q14c

2 marks

Use the results from Questions 14a) and 14b) to evaluate the reasonableness of Khadija's suggestion.

By combining the results, the sample size is increased and the confidence interval width is reduced.
The new sample statistic provides a better estimate for the population parameter.

Marking Criteria

Descriptor	Marks
identifies changed width of confidence interval	1
evaluates the reasonableness of Khadija’s suggestion	1

Q14d

2 marks

The proportion of all Brisbane pet owners who celebrate their pet's birthday is 0.24.

Using the normal approximation, determine the probability that in a randomly selected sample of size 200, more than 30% of pet owners celebrate their pet's birthday.

Using approximation to the normal distribution
Mean $= 0.24$
Standard deviation $= \sqrt{\frac{0.24 \times 0.76}{200}} = 0.0302$
Using GDC
$P(\hat{p} > 0.30) = 0.0235$

Marking Criteria

Descriptor	Marks
correctly determines the mean and standard deviation of the normal distribution	1
determines the probability	1

Q13

2020

QCAA

Paper 2

6 marks

Q13

An online retailer claims that 90% of all orders are shipped within 12 hours of being received. On a particular day, 121 orders were received and 102 orders were shipped within 12 hours.

Q13a

1 mark

State the sample proportion of orders shipped within 12 hours.

Descriptor	Marks
Correctly determines the sample proportion	1

Q13b

3 marks

The distribution of the sample proportion of all orders that are shipped within 12 hours of being received on any day is approximately normal.

Assuming the online retailer's claim is true, find the probability that, in a random sample of 121, less than 85% of all orders are shipped within 12 hours.

Using normal distribution
$\mu = 0.9$
$\sigma = \sqrt{\frac{0.9 \times 0.1}{121}}$

Using GDC
$P(\hat{p} < 0.85)$
$= 0.0334$

Marking Criteria

Descriptor	Marks
Correctly identifies the mean of the population	1
Determines expression for standard deviation of the normal distribution	1
Determines probability of producing sample proportion or less	1

Q13c

2 marks

Use the result from 13b) to evaluate the reasonableness of the online retailer's claim.

There is 3.34% chance of shipping less than 85% of orders when the population parameter is 0.90.
Observing a sample proportion of 0.843 (or even lower) would have occurred by chance less than 3.34% of the time if the retailer's claim is true. Therefore, we suspect that the retailer's claim is dubious.

Marking Criteria

Descriptor	Marks
Identifies meaning of the probability of producing the sample proportion	1
Evaluates reasonableness of the retailer's claim	1

QCAA Mathematical Methods Sampling and proportions

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