Question

1. for a standard normal distribution (z), answer the following:

a. fill in the blanks.
b. p(z > 1.24)
c. p(z < 2.13)
d. p(-0.45 < z < 2.29)
e. find c if p(z > c) = 0.39.
f. find c if p(-c < z < c) = 0.83.

Explanation:

Step1: Recall standard normal distribution

A standard normal distribution $Z$ has mean $\mu = 0$ and standard - deviation $\sigma=1$, so $Z\sim N(0,1)$.

Step2: Calculate $P(Z > 1.24)$

We know that the total area under the standard - normal curve is 1. Using the property $P(Z>z)=1 - P(Z\leq z)$. From the standard normal table, $P(Z\leq1.24) = 0.8925$. So $P(Z > 1.24)=1 - 0.8925=0.1075$.

Step3: Calculate $P(Z < 2.13)$

Looking up the value in the standard normal table, $P(Z < 2.13)=0.9834$.

Step4: Calculate $P(-0.45 < Z < 2.29)$

Using the property $P(a < Z < b)=P(Z < b)-P(Z < a)$. From the standard normal table, $P(Z < 2.29)=0.9890$ and $P(Z < - 0.45)=1 - P(Z < 0.45)=1 - 0.6736 = 0.3264$. So $P(-0.45 < Z < 2.29)=0.9890-0.3264 = 0.6626$.

Step5: Find $c$ when $P(Z > c)=0.39$

Since $P(Z > c)=0.39$, then $P(Z\leq c)=1 - 0.39 = 0.61$. Looking up the value 0.61 in the standard - normal table, $c\approx0.28$.

Step6: Find $c$ when $P(-c < Z < c)=0.83$

We know that $P(-c < Z < c)=2P(Z < c)-1$. So $2P(Z < c)-1 = 0.83$, then $2P(Z < c)=1 + 0.83=1.83$, and $P(Z < c)=0.915$. Looking up the value 0.915 in the standard - normal table, $c\approx1.37$.

Answer:

a. $Z\sim N(0,1)$
b. $0.1075$
c. $0.9834$
d. $0.6626$
e. $c\approx0.28$
f. $c\approx1.37$