91Ó°ÊÓ

Find the following $z-$scores.

a. $Z_{0.03}$

b.$z_{0.005}$

Briefly, for a normally distributed variable, how do you obtain the percentage of all possible observations that lie within a specified range?

State the empirical rule as specialized to variables.

A variable is normally distributed with a mean $6$and standard deviation $2$. Find the percentage of all possible values of the variable that

a. lie between $1$and $7$

b. exceed $5$

c. are less than $4$.

A variable is normally distributed with a mean $10$ and standard deviation$3$. Find the percentage of all possible values of the variable that

a. lie between $6$and $7$.

b. are at least $10$ .

c. are at most $17.5$ .

A variable is normally distributed with mean $6$and standard deviation $2$.

Part (a): Determine and interpret the quartiles of the variable.

Part (b) Obtain and interpret the $85th$percentile.

Part (c) Find the value that $65\%$of all possible values of the variable exceed.

Part (d) Find the two values that divide the area under the corresponding normal curve into a middle area of $0.95$and two outside areas of $0.025$. Interpret your answer.

A variable is normally distributed with mean $10$and standard deviation $3$.

Part (a): Determine and interpret the quartiles of the variable.

Part (b): Obtain and interpret the seventh percentile.

Part (c): Find the value that $35\%$of all possible values of the variable exceed.

Part (d): Find the two values that divide the area under the corresponding normal curve into a middle area of $0.99$and two outside areas of $0.005$. Interpret your answer.

A variable is normally distributed with mean $10$ and standard deviation $3$.

a. Determine and interpret the quartiles of the variable.

b. Obtain and interpret the seventh decile.

c. Find the value that $35\%$ of all possible values of the variable exceed.

d. Find the two values that divide the area under the corresponding normal curve into a middle area of $0.99$ and two outside areas of $0.005$. Interpret your answer.

Giant Tarantulas. One of the larger species of tarantulas is the Grammostola mollicoma, whose common name is the Brazilian giant tawny red. A tarantula has two body parts. The anterior part of the body is covered above by a shell, or carapace. From a recent article by F. Costa and F. Perez-Miles titled "Reproductive Biology of Uruguayan Theraphosids" (The Journal of A rachutology, Vol. 30 , No. 3, pp. 571-587), we find that the carapace length of the adult male $G$. mollicoma is normally distributed with mean $18.14\mathrm{mm}$ and standard deviation $1.76\mathrm{mm}$.

a. Find the percentage of adult male $G$. mollicoma that have carapace length between $16\mathrm{mm}$ and $17\mathrm{mm}$.

b. Find the percentage of adult male $G$. mollicoma that have carapace length cxcceding $19\mathrm{mm}$.

c. Determine and interpret the quartiles for carapace length of the adult male $G$. mollicoma.

d. Obtain and interpret the $95th$ percentile for carapace length of the adult male $G$. mollicoma.

As reported in Runner's World magazine, the times of the finishers in the New York City $10$-km run are normally distributed with mean $61$minutes and standard deviation 9 minutes.

Part (a): Determine the percentage of finishers who have times between $50$and $70$minutes.

Part (b): Determine the percentage of finishers who have times less than $75$minutes.

Part (c): Obtain and interpret the $40$th percentile for the finishing times.

Part (d): Find and interpret the $8$th decile for the finishing times.