In a school, the IQ scores of the students have a normal distribution with a mean value of 100, and a standard deviation of 10. What is the probability that the IQ score of a randomly selected student is at least 120? (10 p)
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To find the probability that the IQ score of a randomly selected student is at least 120, we need to calculate the area under the normal distribution curve to the right of the value 120.
We can use the Z-score formula to convert the IQ score into a standard score (Z-score). The Z-score tells us how many standard deviations an IQ score is away from the mean.
The formula to calculate the Z-score is:
Z = (X - μ) / σ
Where:
Z = Z-score
X = IQ score
μ = Mean of the distribution
σ = Standard deviation of the distribution
In this case, X = 120, μ = 100, and σ = 10.
Plugging in the values, we get:
Z = (120 - 100) / 10
Z = 20 / 10
Z = 2
Now, we need to find the area under the normal distribution curve to the right of Z = 2. This represents the probability that an IQ score is at least 120.
Using a standard normal distribution table or a statistical software, we can find the probability associated with Z = 2. We can also use the cumulative distribution function (CDF) of the normal distribution to find this probability.
Using a standard normal distribution table, the probability associated with Z = 2 is approximately 0.9772.
Therefore, the probability that the IQ score of a randomly selected student is at least 120 is approximately 0.9772 or 97.72%.