For standard normal random variable Z, determine the value of constant c which makes the probability statements given below correct

Kody Whitaker 2022-09-27 Answered
For standard normal random variable Z, determine the value of constant c which makes the probability statements given below correct
a) Φ ( c ) = 0.8888
b) P ( 0 Z c ) = 0.334
c) P ( c Z c ) = 0.488
e) P ( c | Z | ) = 0.154
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Answers (2)

Zackary Galloway
Answered 2022-09-28 Author has 17 answers
Step 1
a) Φ ( c ) = 0.8888 P ( Z c ) = 0.8888
From standard norma tables, the value corresponding to 0.8888 cumulative probability is 1.22
Hence, c = 1.22
Step 2
b) P ( 0 Z c ) = 0.334 P ( Z c ) P ( Z 0 ) = 0.334 P ( Z c ) 0.5 = 0.334 ( since   P ( Z 0 ) = P ( Z 0 ) = 0.5 ) P ( Z c ) = 0.834
From standard normal tables, the value corresponding to 0.8888 cumulative probability is 0.97
Hence, c = 0.97
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Kody Whitaker
Answered 2022-09-29 Author has 1 answers
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c) P ( c Z ) = 0.271 P ( Z c ) = 0.271 1 P ( Z < c ) = 0.271 P ( Z < c ) = 1 0.271 P ( Z < c ) = 0.729
From standard norma tables, the value corresponding to 0.729 cumulative probability is 0.61
Hence, c = 0.61
d) P ( c Z c ) = 0.488 P ( Z c ) P ( Z c ) = 0.488 P ( Z c ) P ( Z > c ) = 0.488   (Since Z follows normal distribution which is symmetric) P ( Z c ) [ 1 P ( Z c ) ] = 0.488 2 P ( Z c ) 1 = 0.488 2 P ( Z c ) = 1.488 P ( Z c ) = 0.744From standard normal tables, the value corresponding to 0.744 cumulative probability is 0.66
Hence, c = 0.66
e) P ( x | Z | ) = 0.154 P ( c Z c ) = 0.154 ⇒⇒ P ( Z c ) P ( Z c ) = 0.154 P ( Z c ) P ( Z > c ) = 0.154   (Since Z follows normal distribution which is symmetric) P ( Z c ) [ 1 P ( Z c ) ] = 0.154 2 P ( Z c ) 1 = 0.154 2 P ( Z c ) = 1.154 P ( Z c ) = 0.577
From standard normal tables, the value corresponding to 0.577 cumulative probability is 0.194
Hence, c = 0.19
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