In: Statistics and Probability
If Z is Standard Normal Variable, answer the following questions. Please show you complete work for each question.
a. What is the Z value if the area under the curve and to the right of Z is 0.95
b. What is the probability of Z is within the interval of -1.75 and 2.61?
c. What are the middle Z values (-Z and + Z) such that the area under the probability curve from -Z to +Z is 0.97?
(a) If Area to the right is 0.95, then area to the left is 0.05, i.e P(X < x) = 0.05
The Z value at this probability = -1.645
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(b) The probability = P(Z = 2.61) - P(Z = -1.75)
The probability at Z = 2.61 is = 0.9955 and The probability at Z = -1.75 is = 0.0401
Therefore the required probability = 0.9955 - 0.0401 = 0.9554
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(c) Since the Z values are symmetrical (Same in value), therefore 2 * P(Z) - 1 = 0.97
Therefore 2*P(Z) = 1.97
P(Z) = 1.97 / 2 = 0.985
The z value at p = 0.985 is 2.17
Therefore the middle values are Z = -2.17 and Z = +2.17
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