The probability P(X < 105), assuming that the random variable X is normally distributed, with mean μ = 90 and standard deviation σ = 12 is calculated to be 0.8944.
When a random variable X is normally distributed, with the mean as μ, and standard deviation as σ, then the probability P(X < x) is calculated using the Z-score table, as the probability P(Z < z), where z is calculated using the formula, z = (x - μ)/σ.
In the question, we are asked to find the probability P(X < 105), assuming that the random variable X is normally distributed, with mean μ = 90 and standard deviation σ = 12.
We calculate as follows:
P(X < 105)
= P(Z < (105-90)/12)
= P(Z < 1.25)
Now we check the z-score table for the value of z as 1.25.
On the left column we choose the value 1.2.
On the top row, we choose the value .05.
The intersection of these, gives us the value,
P(Z < 1.25) = 0.8944.
Thus, the probability P(X < 105), assuming that the random variable X is normally distributed, with mean μ = 90 and standard deviation σ = 12 is calculated to be 0.8944.
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