SAC 3 HOMEWORK PREPARATION – COMPONENT 0
Contents covered: Linear combinations of random variables and hypothesis testing.
BACKGROUND INFORMATION ON CANE TOADS
https://australian.museum/blog/amri-news/canetoads-mammal-declines/ (July 2022)
Cane toads were introduced in Australia, in North Queensland, in 1935, in order to control the cane beetle
which destroyed sugar cane crops. Cane toads are native to South and Central America. In Australia, they
have no predators and are a threat to native species. They are hardy and adaptable, and it is understood that
their numbers grew from 102 to an estimated 200 million in 2020. They are an invasive species which
destroys the native ecosystem, typically devastating local native predators by 90% within a few months of
arrival, and their habitat expands at a rate of around 50km per year.
For more information, refer to
•
https://www.wwf.org.au/news/blogs/10-facts-about-cane-toads#gs.xpptpb
•
https://www.dcceew.gov.au/environment/invasive-species/feral-animals-australia/cane-toads
•
https://www.biorxiv.org/content/10.1101/344796v1.full
•
https://pestsmart.org.au/toolkit-resource/methods-for-the-field-euthanasia-of-cane-toads/
•
https://www.canetoadsinoz.com/
https://www.business.qld.gov.au/industries/farms-fishing-forestry/agriculture/landmanagement/health-pests-weeds-diseases/pests/invasive-animals/other/cane-toad
The numbers used in this SAC try to be realistic but should not be considered a true reflection of the actual
numbers, which differ according to location and time.
MLC – 12 SM – UNIT 4 – SAC 3 -2022 – COMPONENT 0
1
PROBABILITY DENSITY FUNCTION; MEASURE OF CENTRE AND OF SPREAD.
A random variable X has a probability density function given by
1
2
𝑓𝑓(𝑥𝑥) = �9 (4𝑥𝑥 − 𝑥𝑥 ) 𝑖𝑖𝑖𝑖 0 < 𝑥𝑥 < 3
0
𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
a) Find its expected value (=mean)
b) Find its variance.
c) Find its standard deviation.
d) Find Pr(X≤ 2)
MLC – 12 SM – UNIT 4 – SAC 3 -2022 – COMPONENT 0
2
LINEAR COMBINATION OF RANDOM VARIABLES
1
1. Let 𝑋𝑋� be such that 𝑋𝑋� = 𝑛𝑛 (𝑋𝑋1 + 𝑋𝑋2 + ⋯ . 𝑋𝑋𝑛𝑛 ) where the random variable 𝑋𝑋1 is distributed with mean μ
and variance 𝜎𝜎 2 and the random variables 𝑋𝑋2 , … 𝑋𝑋𝑛𝑛 are independent and identically distributed to
𝑋𝑋1 .
a) Using properties of the expected value E, show that E(𝑋𝑋�)=μ for all 𝑛𝑛∈ℕ
b) Find an expression for Var(𝑋𝑋�) in terms of σ and 𝑛𝑛.
c) Hence, find an expression for the standard deviation of 𝑋𝑋�, 𝑠𝑠𝑠𝑠(𝑋𝑋�)
MLC – 12 SM – UNIT 4 – SAC 3 -2022 – COMPONENT 0
3
2. The notation 𝑋𝑋~𝑁𝑁(0,9) means that 𝑋𝑋 is normally distributed where E(X)=0 and Var(X)=9.
If 𝑌𝑌~𝑁𝑁(4,1.5), describe the distribution of 2X+5Y.
CONFIDENCE INTERVALS
1.
Let X be a random variable with mean μ and standard deviation σ, and 𝑋𝑋� the sample mean.
The central limit theorem states that, for large 𝑛𝑛, 𝑋𝑋� is approximately normal with mean μ and standard
deviation
𝜎𝜎
√𝑛𝑛
𝜎𝜎
2
, that is 𝑋𝑋�~𝑁𝑁(μ, � � ). Using this in conjunction with the fact that Pr(−1.96 ≤ 𝑍𝑍 ≤
√𝑛𝑛
1.96) = 0.95, derive the expression for the 95% confidence interval (𝑥𝑥̅ − 1.96
MLC – 12 SM – UNIT 4 – SAC 3 -2022 – COMPONENT 0
𝜎𝜎
√𝑛𝑛
, 𝑥𝑥̅ + 1.96
𝜎𝜎
√𝑛𝑛
)
4
2.
After surveying 30 swimmers as they entered the local sports and aquatic centre, it was found that
the average distance they intended to swim was 1.2 km. The sample standard deviation was 0.5km.
a) Estimate the average distance people were intending to swim that day.
b) Find a 95% confidence interval for your estimate.
c) What sample size would be needed to reduce the interval to ±0.1km at the 95% level of
confidence.
MLC – 12 SM – UNIT 4 – SAC 3 -2022 – COMPONENT 0
5
d) Discuss the effect of the sample size on the confidence interval for a 95% level of confidence.
Use mathematical language, notation and/or method to explain and describe the changes
observed.
e) Discuss how the level of confidence impacts the confidence interval. Use mathematical language,
notation and/or method to explain and describe the changes observed.
MLC – 12 SM – UNIT 4 – SAC 3 -2022 – COMPONENT 0
6
HYPOTHESES TESTING
The drying time for a particular type of paint is known to be normally distributed with μ=75minutes and
σ=9.4 minutes. A new additive has been developed that is supposed to change the drying time. After testing
100 samples, the observed drying time is 𝑥𝑥̅ = 71.2 minutes. We are investigating whether the new additive
changes the drying time or not.
a) State the null and alternative hypotheses.
b) Test the claim at the 1% level of significance.
c) If 𝑥𝑥̅ = 72.9, what is the conclusion , using 𝛼𝛼 = 0.01?
MLC – 12 SM – UNIT 4 – SAC 3 -2022 – COMPONENT 0
7
d) What is the 𝑝𝑝 −value if 𝑥𝑥̅ = 72.9? What would the conclusion be at a 1% level of significance?
e) What is the significance level α for the test procedure that rejects 𝐻𝐻0
when 𝑍𝑍 < −2.88 or 𝑍𝑍 > 2.88?
MLC – 12 SM – UNIT 4 – SAC 3 -2022 – COMPONENT 0
8
f) Give an average observed drying time from 100 samples, that would lead to 𝐻𝐻0 being rejected, using
this level of confidence (as determined in e) above).
MLC – 12 SM – UNIT 4 – SAC 3 -2022 – COMPONENT 0
9