A random variable is normally distributed with a mean of 0 and a standard deviation of 1 - HSC - SSCE Mathematics Advanced - Question 23 - 2023 - Paper 1
Question 23
A random variable is normally distributed with a mean of 0 and a standard deviation of 1. The table gives the probability that this random variable lies below z for ... show full transcript
Worked Solution & Example Answer:A random variable is normally distributed with a mean of 0 and a standard deviation of 1 - HSC - SSCE Mathematics Advanced - Question 23 - 2023 - Paper 1
Step 1
Calculate the z value
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Answer
To find the z-value for 11.93 kg, use the formula:
z=σx−μ
Where:
x = 11.93 kg
μ = 10.40 kg
σ = 1.15 kg
Substituting the values:
z=1.1511.93−10.40=1.151.53≈1.33(to two decimal places).
Step 2
Find the probability from the table
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Answer
Using the z-value calculated, we refer to the provided table to find:
The probability corresponding to z = 1.33 is 0.9082.
Therefore, the probability of a koala weighing less than 11.93 kg is 0.9082.
Step 3
Calculate probability of weighing more than 11.93 kg
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Answer
To find the probability of a koala weighing more than 11.93 kg, we calculate:
P(more than 11.93)=1−P(less than 11.93)=1−0.9082=0.0918.
Step 4
Calculate expected number of koalas
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In a group of 400 adult male koalas, the expected number of koalas weighing more than 11.93 kg can be found:
Number of koalas=P(more than 11.93)×400=0.0918×400=36.72.
Thus, we would expect approximately 37 koalas (rounding can accept either 36 or 37).
