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 for 11.93 kg
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Answer
To find the z-score, use the formula:
z = \frac{x - \mu}{\sigma}$$
Substituting the values:
z = \frac{11.93 - 10.40}{1.15} = \frac{1.53}{1.15} \approx 1.33$$
Thus, the calculated z-value is approximately 1.33.
Step 2
Find the probability from the table for z = 1.33
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Answer
From the table, the probability of a z-value of 1.33 is 0.9082.
Step 3
Calculate the probability of weighing more than 11.93 kg
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Answer
To find the probability of weighing more than 11.93 kg, use:
P(more than 11.93)=1−P(Z≤1.33)
Substituting the probability:
P(more than 11.93)=1−0.9082=0.0918
Step 4
Calculate the expected number of koalas that weigh more than 11.93 kg
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Now, multiply the probability by the total number of koalas:
Number of koalas=0.0918×400=36.72
Rounded, we expect approximately 37 koalas to weigh more than 11.93 kg, but 36 is also an acceptable answer.
