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10. Samples and estimation
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AustraliaVIC
VCE 12 Methods 2023

10.04 Confidence intervals

Lesson
Worksheet
Practice

Interactive practice questions

Given a sample size of $530$530 and a sample proportion of $40%$40%, find the approximate two-sided $95%$95% confidence interval for the population proportion using the given table.

Give your answer in the form $\left(a,b\right)$(a,b), and round your answer to two decimal places.

Easy
< 1min

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$500 people who work in the city if they use public transport to commute to work and $72.5%$72.5% of them responded that they did.

Easy
< 1min

Ten samples, each of size $150$150, have their two-sidedΒ $90%$90% confidence interval calculated. How many of these samples would we expect to contain the true population proportion?

Easy
< 1min

One hundred samples, each of size $300$300, have their two-sidedΒ $95%$95% confidence interval calculated. How many of these samples would we expect to contain the true population proportion?

Easy
< 1min
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Outcomes

U34.AoS4.4

statistical inference, including definition and distribution of sample proportions, simulations and confidence intervals: - distinction between a population parameter and a sample statistic and the use of the sample statistic to estimate the population parameter - simulation of random sampling, for a variety of values of 𝑝 and a range of sample sizes, to illustrate the distribution of 𝑃^ and variations in confidence intervals between samples - concept of the sample proportion as a random variable whose value varies between samples, where 𝑋 is a binomial random variable which is associated with the number of items that have a particular characteristic and 𝑛 is the sample size - approximate normality of the distribution of P^ for large samples and, for such a situation, the mean 𝑝 (the population proportion) and standard deviation - determination and interpretation of, from a large sample, an approximate confidence interval for a population proportion where 𝑧 is the appropriate quantile for the standard normal distribution, in particular the 95% confidence interval as an example of such an interval where 𝑧 β‰ˆ 1.96 (the term standard error may be used but is not required).

U34.AoS4.8

the concept of confidence intervals for proportions, variation in confidence intervals between samples and confidence intervals for estimates

U34.AoS4.12

simulate repeated random sampling and interpret the results, for a variety of population proportions and a range of sample sizes, to illustrate the distribution of sample proportions and variations in confidence intervals

U34.AoS4.13

calculate sample proportions and approximate confidence intervals for population proportions

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