 # 10.04 Confidence intervals for sample proportions

Worksheet
Confidence intervals
1

Given a sample size of 530 and a sample proportion of 40\%, find the approximate two-sided 95\% confidence interval for the population proportion. Round your values to two decimal places.

2

To assist with road and transport planning, the council wants to estimate the proportion of people who use public transport to commute to work in the city. They asked a sample of 500 people who work in the city if they use public transport to commute to work and 72.5\% of them responded that they did.

Determine the 90\% confidence interval for the true proportion of people who use public transport to commute to work in the city. Round your values to two decimal places.

3

For each of the following scenarios, find the number of samples expected to contain the true population proportion:

a

Ten samples, each of size 150, have their two-sided 90 \% confidence interval calculated.

b

One hundred samples, each of size 300, have their two-sided 95 \% confidence interval calculated.

c

500 samples, each of size 250, have their two-sided 99 \% confidence interval calculated.

d

k samples, each of size n, have their two-sided x \% confidence interval calculated.

4

Ten samples, each of size 150, have their two-sided 90 \% confidence interval calculated. Which of the following diagrams of confidence intervals best matches our expectation of how many should contain the true population proportion?

A
B
C
D
5

Several samples have their two-sided 80 \% confidence interval calculated. Which of the following diagrams of confidence intervals best matches our expectation of how many should contain the true population proportion?

A
B
C
D
6

A sample of size 160 is taken from the population, and the sample proportion is found to be 0.48.

a

Calculate the approximate two-sided 95\% confidence interval for the true proportion. Round your values to two decimal places.

b

How much confidence can we have that the true proportion lies within the interval in part (a)?

c

State the probability that the true proportion lies within the interval in part (a).

7

A sample of size 170 is taken from the population, and the sample proportion is found to be 0.3.

Calculate the approximate two-sided 99\% confidence interval for the true proportion. Round your values to two decimal places.

8

A sample of size 80 is taken from the population, and the sample proportion is found to be 0.69.

Calculate the approximate two-sided 85\% confidence interval for the true proportion. Round your values to two decimal places.

9

In a sample of 30 students from a school, 9 of them would prefer different school hours.

Calculate the approximate two-sided 90\% confidence interval for the probability of a student at the school preferring different school hours. Round your values to two decimal places.

10

When a major chocolate manufacturer created a new chocolate bar, they gathered a sample of 500 people to taste test the bar and give their feedback. 125 of them said they would buy the bar.

a

Find the approximate two-sided 95\% confidence interval for the true proportion of people who would buy the chocolate bar. Round your values to two decimal places.

b

11

Marine biologists studied a sample of 40 whales in the Atlantic Ocean and found that 11 of them had elevated levels of metals in their blood.

a

Find the approximate two-sided 99\% confidence interval for the probability of a whale having elevated levels of mercury in its blood. Give your answer in the form \left(a, b\right), and round your values to two decimal places.

b

12

In a study of 60 households that have a mortgage, 50\% of them were found to be under mortgage stress.

Hence, find the 85\% confidence interval for the proportion of mortgaged households that are under mortgage stress. Round your values to two decimal places.

13

A two-sided 95\% confidence interval is calculated for a sample proportion \hat{p}, and \left(0.218, 0.382\right) is the confidence interval.

a

Find \hat{p}.

b

Given that the sample size was 120, how many in the sample exhibited the particular characteristic being observed?

14

Three samples from the same population were taken and a particular characteristic was observed:

• Sample A: The characteristic appeared 48 times out of 70.

• Sample B: The characteristic appeared 126 times out of 210.

• Sample C: The characteristic appeared 80 times out of 120

a

Calculate the approximate two-sided 90 \% confidence interval for the following samples to two decimal places:

i

Sample A

ii

Sample B

iii

Sample C

b

Which sample has the smallest confidence interval width?

c

Which samples have their lower and upper bounds more closely aligned with each other?

15

A two-sided 99.5\% confidence interval is calculated for a sample proportion \hat{p}, and \left(0.279, 0.846\right) is the confidence interval. Determine \hat{p}.

16

Find the sample proportion \hat{p} for the two-sided confidence interval \left(0.652, 0.711\right).

Margin of error
17

Find the margin of error for the confidence interval \left(0.311, 0.723\right).

18

Find the margin of error for each of the following, rounding your answers to three decimal places:

a

A two-sided 99\% confidence interval is calculated for a sample proportion \hat{p} and \left(0.672, 0.841\right) is the confidence interval.

b

An approximate two-sided 80\% confidence interval for a sample of size 120 that has a sample proportion of 0.6.

c

An approximate two-sided 99\% confidence interval for a sample of size 90 that has a sample proportion of 0.3.

d

An 85\% confidence interval for a sample of size 70 that has a sample proportion of 0.62.

19

Two students performed a similar simulation on their calculator and a particular characteristic was observed:

• Simulation 1 : The characteristic appeared 37 times out of 100.

• Simulation 2 : The characteristic appeared 63 times out of 180.

a

Calculate the approximate two-sided 95 \% confidence interval for the following simulations to two decimal places:

i

Simulation 1

ii

Simulation 2

b

Which simulation's confidence interval has the largest margin of error?

20

A survey of 1700 voters commissioned by a political party found that 46.5\%, on a two-party preferred basis, intended to vote for the party. Find the approximate margin of error with a two-sided 90\% confidence interval, correct to three decimal places.

21

A trout farm will only harvest fish if they are at least 41 \text{ cm} long. 600 trout are caught and measured, and 75\% were found to be at least the minimum length. Find the approximate margin of error for a two-sided 95\% confidence interval. Give your answer as a percentage correct to two decimal places.

22

For each of the following:

i

Calculate \hat p.

ii

Calculate the margin of error.

iii

State the z-score associated with the margin of error, correct to two decimal places.

iv

Calculate the size of the sample, n.

v

State the amount in the sample who exhibited the particular characteristic being observed, to the nearest whole number.

a

A two-sided 90\% confidence interval is calculated for a sample proportion \hat{p}, and \left(0.638, 0.762\right) is the confidence interval.

b

A two-sided 95\% confidence interval is calculated for a sample proportion \hat{p}, and \left(0.516, 0.684\right) is the confidence interval.

c

A two-sided 99\% confidence interval is calculated for a sample proportion \hat{p}, and \left(0.382, 0.618\right) is the confidence interval.

d

A two-sided 85\% confidence interval is calculated for a sample proportion \hat{p}, and \left(0.333, 0.467\right) is the confidence interval.

23

For each of the following surveys:

i

Calculate the sample proportion, \hat p.

ii

Calculate the margin of error.

iii

Calculate the sample size, n, to the nearest whole number.

a

A survey is carried out and the approximate two-sided 95\% confidence interval for the population proportion is \left(0.57, 0.77\right).

b

A survey is carried out and the approximate two-sided 99.5\% confidence interval for the population proportion is \left(0.41, 0.85\right).

c

A survey is carried out and the approximate two-sided 90\% confidence interval for the population proportion is \left(0.36, 0.71\right).

24

True or False: The smaller the margin of error, the smaller the confidence interval.

25

For each of the following, state which has the greatest margin of error:

a
• A two-sided 90 \% confidence interval.

• A two-sided 95 \% confidence interval.

b
• A two-sided 95 \% confidence interval.

• A two-sided 99 \% confidence interval.

26

For the each of the following, state which sample will have the smaller margin of error:

a

Two samples, A and B, have the same \hat{p}. Each sample has a two-sided 95 \% confidence calculated. Sample A is of size 250. Sample B is of size 100.

b

Two samples, A and B, are the same size. Each sample has a 99 \% confidence calculated. \hat{p} for Sample A is 0.4 and \hat{p} for Sample B is 0.1.

27

For the each of the following, state which sample will have the larger margin of error:

a

Two samples, A and B, have the same \hat{p}. Each sample has a two-sided 90 \% confidence calculated. Sample A is of size 100. Sample B is of size 500.

b

Two samples, A and B, are the same size. Each sample has a 90\% confidence calculated. \hat{p} for Sample A is 0.2 and \hat{p} for Sample B is 0.3.

Minimum sample size
28

For each scenario, use technology to find the minimum value of n, the sample size:

a

A survey is to be carried out with the aim of having the approximate two-sided 90\% confidence interval on the population proportion with margin of error being less than 0.085. It is known that the sample proportion was 0.7.

b

A survey is to be carried out with the aim of having the approximate two-sided 95\% confidence interval on the population proportion with margin of error no more than \\ 0.085. It is known that the sample proportion was 0.6.

c

A survey is to be carried out with the aim of having the approximate two-sided 99\% confidence interval on the population proportion with margin of error no more than \\ 0.045. It is known that the sample proportion was 0.7.

29

Calculate the minimum size of the sample required to achieve the following:

a

A margin of error less than 3 \% in an approximate 90 \% confidence interval when \hat{p}=0.7.

b

A margin of error less than 0.45 \% in an approximate 95 \% confidence interval when \\ \hat{p}=0.37.

30

Use technology to determine the minimum sample size required to achieve a margin of error of 3\% in an approximate two-sided 95\% confidence interval for the proportion p of primary school children in Australia who play competitive sport. The sample proportion \hat{p} is found to be 0.6.

31

The Widget and Trinket Emporium wishes to conduct a quality control survey on all its widgets, with the aim of having a margin of error no more than 1\% in an approximate two-sided 90\% confidence interval. Given that the sample proportion of compliant widgets was 0.9, use technology to find the minimum value of n, the sample size.

32

A survey is carried out with a sample size of 175 and found that 42 participants display a given attribute.

a

Find the approximate two-sided 90 \% confidence interval. Round your values to two decimal places.

b

Determine the minimum size of the sample required to have a 90\% confidence interval of width less than 0.35 if \hat{p}=0.25.

33

A survey is carried out with a sample size of 230 and found that 155 participants display a given attribute.

a

Find the approximate two-sided 87 \% confidence interval. Round your values to two decimal places.

b

Determine the minimum size of the sample required to have a 87\% confidence interval of width less than 0.15 if \hat{p}=0.70.

34

A survey is carried out with a sample size of 200 and found that 80 participants display a given attribute.

a

Find the approximate two-sided 95 \% confidence interval. Round your values to two decimal places.

b

If we wish to have a confidence interval with a width of less than 0.1, how many people would need to be surveyed if the sample proportion was to be approximately the same?

c

If a sample larger than 200 cannot be obtained but a confidence interval of width less than 0.1 is to be created, what level of confidence could be claimed for this interval with the same sample proportion? Give your answer as a whole number percentage.

### Outcomes

#### 4.5.3.1

understand the concept of an interval estimate for a parameter associated with a random variable

#### 4.5.3.2

use the approximate confidence interval [ ˆp-√(ˆp(1−ˆp)/n, ˆp+z√(ˆp(1−ˆp)/n), as an interval estimate for p, where z is the appropriate quantile for the standard normal distribution

#### 4.5.3.3

define the approximate margin of error E=z√(ˆp (1−ˆp)/n and understand the trade-off between margin of error and level of confidence