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Answer:
a) P(X = 111) = 0; P(X < 111) = 0.5987; P(X ≤ 111) = 0.5987; b) 0.0124; No
Step-by-step explanation:
For part a, we use z scores. The formula for the z score of an individual value is
Our mean, μ, is 110 and our standard deviation, σ, is 4.
The probability that x is equal to a given value is 0, regardless of the value, the mean or the standard deviation. This is because the probability in a normal distribution is the area under the curve; this means there must be a range of numbers.
To find P(X < 111),
z = (111-110)/4 = 1/4 = 0.25
Using a z table, we see that the area under the curve to the right of this is 0.5987.
For P(X ≤ 111), we use the same probability as P(X < 111). There is no distinction between "less than" and "less than or equal to."
For part b,
2.5 standard deviations above the mean is z = 2.5. 2.5 standard deviations below the mean is z = -2.5.
Using the z table, the area under the curve to the right of z = 2.5 is 0.9938. The area under the curve to the right of z = -2.5 is 0.0062. This makes the area between them
0.9938-0.0062 = 0.9876. This means anything farther from the mean than this is
1-0.9876 = 0.0124. This does not have anything to do with the value of the mean or the standard deviation; this is because we already have the z score.