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

Kody Whitaker

Kody Whitaker

Answered question

2022-09-27

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

Answer & Explanation

Zackary Galloway

Zackary Galloway

Beginner2022-09-28Added 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
Kody Whitaker

Kody Whitaker

Beginner2022-09-29Added 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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