Question 701179: Assume that the mean SAT score in Mathematics for 11th graders across the nation is 500, and that the standard deviation is 100 points. Find the probability that the mean SAT score for a randomly selected group of 150 11th graders is 500.
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Assume that the mean SAT score in Mathematics for 11th graders across the nation is 500, and that the standard deviation is 100 points. Find the probability that the mean SAT score for a randomly selected group of 150 11th graders is 500.
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Note: The probability of ANY particular value in a continuous distribution
is always zero.
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Your Problem:
By the Central Limit Theorem
The mean of the sample means is 500
The std of the sample means is 100/sqrt(150) = 8.1649
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BUT, look what happens when you find the z-score.
z(500) = (500-100/8.1649 = 48.99
AND
P(x-bar = 500) = P(z = 48.99) = ZERO
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Why ZERO?
Probability for a continuous distribution is modeled by
area ABOVE AN INTERVAL and below a normal curve.
But there is no INTERVAL when a single Z-value is the base.
The area ABOVE is zero because the base is ZERO.
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In general, the probability of ANY particular value is ZERO
in a continuous distribution.
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Cheers,
Stan H.
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