Question

a. the area under the normal curve between ±z is 84%. find z (enter a positive value). answer: z = b. the area under the normal curve to the left of z is 84%. find z. answer: z =

Explanation:

Step1: Use z - table (standard normal table)

We know that the standard normal table gives the cumulative - distribution function values for the standard normal distribution $N(0,1)$. The cumulative distribution function $\varPhi(z)$ gives the area under the normal curve to the left of $z$.

Step2: Find the z - value for part a

If the area between $\pm z$ is $84\%$ or $0.84$, then the area in the two tails is $1 - 0.84=0.16$. The area in each tail is $\frac{0.16}{2}=0.08$. The area to the left of $-z$ is $0.08$ and the area to the left of $z$ is $1 - 0.08 = 0.92$. Looking up $0.92$ in the standard normal table, we find that $z\approx1.41$.

Step3: Find the z - value for part b

If the area to the left of $z$ is $84\%$ or $0.84$, we look up $0.84$ in the standard normal table. The $z$ - value corresponding to an area of $0.84$ is approximately $0.99$.

Answer:

a. $z = 1.41$
b. $z = 0.99$