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10. Samples and estimation
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AustraliaVIC
VCE 12 Methods 2023

10.02 Distribution of sample proportions

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Interactive practice questions

From a sample of $400$400 young workers in Seattle, $37$37 had amber eyes.

What is the sample proportion of amber eyes for young workers in Seattle?

Easy
< 1min

A manufacturer of cement trucks checked some of the cement trucks being distributed for faults. On one day, $75$75 cement trucks were checked and $16$16 of these were found to have some kind of fault.

What was the sample proportion for faulty cement trucks?

Easy
< 1min

At an electronics store, cameras are sold at prices ranging from $\$150$$150 to $\$630$$630. On a particular day, there are $106$106 cameras on sale and $25$25 of these cost less than $\$260$$260.

If Irene came into the store and selected a camera at random, estimate the probability that camera would cost less than $\$260$$260.

Easy
< 1min

A survey of $115$115 randomly selected people in Busan found that $6$6 of them were aged over $55$55.

A second survey of $2183$2183 randomly selected people in Busan found that $475$475 of them were aged over $55$55.

Easy
1min
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Outcomes

U34.AoS4.7

the definition of sample proportion as a random variable and key features of the distribution of sample proportions

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 ≈ 1.96 (the term standard error may be used but is not required).

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

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