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9. Continuous random variables
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AustraliaVIC
VCE 12 Methods 2023

9.07 Applications of the normal distribution

Lesson
Worksheet
Practice

Interactive practice questions

The operating times of phone batteries are approximately normally distributed with mean $34$34 hours and a standard deviation of $4$4 hours. Answer the following questions using your calculator:

a

Approximately what percentage of batteries last between $33$33 and $38$38 hours?

Round your answer to the nearest percent.

b

Approximately what percentage of batteries last between $28$28 hours and $41$41 hours?

c

Any battery that lasts less than $23$23 hours is deemed faulty. If a company manufactured $51000$51000 batteries, approximately how many batteries would they be able to sell? Round your answer to the nearest integer.

Medium
1min

The height of sunflowers is approximately normally distributed, with a mean height of $1.6$1.6 m and a standard deviation of $8$8 cm.

Medium
1min

In the Maths Methods course at Winter Heights High, the mean mark for the year was $62%$62% and the standard deviation was $13%$13%.

Medium
3min

The length of the tail of a domestic cat is normally distributed with a mean of $25$25 cm and a standard deviation of $2.2$2.2 cm.

Use the capabilities of your CAS calculator to answer the following, rounding your answers to three decimal places:

Medium
9min
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Outcomes

U34.AoS4.11

apply probability distributions to modelling and solving related problems

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).

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