Answer :
The probability that a randomly selected value from this normally distributed set of data is greater than 25.4 is 0.8413.
In this question, the student is dealing with the Normal Distribution in mathematics, specifically statistics. The question provides the mean and standard deviation of a normally distributed set of data and asks for the probability of a value being greater than a given number.
Firstly, to solve questions about normal distributions, one needs to standardize the value in question to a Z-score, which is a measure of how many standard deviations an element is from the mean. The Z-score is calculated using the following formula:
Z = (X - μ) / σ
Where,
- X is the value in question
- μ is the mean
- σ is the standard deviation
In this case, X=25.4, μ=84.5, and σ=59.1, so we substitute these values into the formula to get:
Z = (25.4 - 84.5) / 59.1 = -1.00
Now, we need to find P(X>25.4), which is the same as finding P(Z > -1.00) in a standard normal distribution. From the Z-table, we know that P(Z< -1.00) = 0.1587
Since in a normal distribution, the total probability equals to 1, so P(Z > -1.00) = 1 - P(Z< -1.00) = 1 - 0.1587 = 0.8413
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