Chapter 2.2, Problem 61E

(a)

To determine

Proportion of time that Mrs. Starnes take to finish an easy Sudoku puzzle in less than 3 minutes.

Answer to Problem 61E

Around 0.0052 of the time does Mrs. Starnes finish an easy Sudoku puzzle in less than 3 minutes.

Explanation of Solution

Given Information:

Time value, $x=3\text{minutes}$

Mean, $\mu =5.3\text{minutes}$

Standard deviation, $\sigma =0.9\text{minute}$

Calculations:

Calculate the z − score,

  $z=\frac{x-\mu }{\sigma }=\frac{3-5.3}{0.9}\approx -2.56$

Use normal probability table in the appendix, to find the corresponding probability.

See the row that starts with -2.5 and the column that starts with .06 of the standard normal probability table for $P\left(z<-2.56\right)$ .

  $P\left(x<3\right)=P\left(z<-2.56\right)=0.0052$

Therefore,

Around 0.0052 of the time does Mrs. Starnes finish an easy Sudoku puzzle in less than 3 minutes.

(b)

To determine

Percentage of time that Mrs. Starnes takes more than 6 minutes to complete an easy puzzle.

Answer to Problem 61E

Around 21.77% of the time Mrs. Starnes takes more than 6 minutes to complete an easy puzzle.

Explanation of Solution

Given Information:

Time value, $x=6\text{minutes}$

Mean, $\mu =5.3\text{minutes}$

Standard deviation, $\sigma =0.9\text{minute}$

Calculations:

Calculate the z − score,

  $z=\frac{x-\mu }{\sigma }=\frac{6-5.3}{0.9}\approx 0.78$

Use normal probability table in the appendix, to find the corresponding probability.

See the row that starts with 0.7 and the column that starts with .08 of the standard normal probability table for $P\left(z<0.78\right)$ .

  $\begin{array}{l}P\left(x>6\right)=P\left(z>0.78\right)\\ =1-P\left(z<0.78\right)\\ =1-0.7823\\ =0.2177\\ =21.77\%\end{array}$

Therefore,

Around 21.77% of the time Mrs. Starnes takes more than 6 minutes to complete an easy puzzle.

(c)

To determine

Percentage of easy Sudoku puzzles completed by Mrs. Starnes between 6 and 8 minutes.

Answer to Problem 61E

Around 21.64% of theeasy Sudoku puzzles take Mrs. Starnes between 6 and 8 minutes to complete.

Explanation of Solution

Given Information:

Time value, $x=6\text{or}8\text{minutes}$

Mean, $\mu =5.3\text{minutes}$

Standard deviation, $\sigma =0.9\text{minute}$

Calculations:

Calculate the z − score,

  $z=\frac{x-\mu }{\sigma }=\frac{6-5.3}{0.9}\approx 0.78$

Or

  $z=\frac{x-\mu }{\sigma }=\frac{8-5.3}{0.9}=3.00$

Use normal probability table in the appendix, to find the corresponding probability.

See the row that starts with 0.7 and the column that starts with .08 of the standard normal probability table for $P\left(z<0.78\right)$ .

Or

See the row that starts with 3.0 and the column that starts with .00 of the standard normal probability table for $P\left(z<3.00\right)$ .

  $\begin{array}{l}P\left(6<x<8\right)=P\left(0.78<z<3.00\right)\\ =P\left(z<3.00\right)-P\left(z<0.78\right)\\ =0.9987-0.7823\\ =0.2164\\ =21.64\%\end{array}$

Therefore,

Around 21.64% of theeasy Sudoku puzzles take Mrs. Starnes between 6 and 8 minutes to complete.