Question

Sudoku Mrs. Starnes enjoys doing Sudoku puzzles. The time she takes to complete an easy puzzle can be modeled by a Normal distribution with a mean of 5.3 minutes and a standard deviation of 0.9 minutes.

a. What proportion of the time does Mrs. Starnes finish an easy Sudoku puzzle in less than 3 minutes?

b. How often does it take Mrs. Starnes more than 6 minutes to complete an easy puzzle?

c. What percent of easy Sudoku puzzles take Mrs. Starnes between 6 and 8 minutes to complete?

Short answer

Part (a) Mrs. Starnes completes an easy Sudoku puzzle in less than $3$minutes around $0.0052$of the time.

Part (b) Mrs. Starnes spends more than $6$minutes on an easy task around $21.77$percent of the time.

Part (c) Mrs. Starnes solves a simple Sudoku puzzle in under $3$ minutes, or $0.0052$ of the time.

Step-by-step solution

Part (a) Step 1: Given information

Time value,$x=3minutes$

Mean,$\mu =5.3minutes$

Standard deviation,$\sigma =0.9minute$

Part (a) Step 2: Concept

The formula used:$z=\frac{x-\mu }{\sigma }$

Part (a) Step 3: Calculation

Calculate the $z-$score,

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

Use the normal probability table in the appendix to get the corresponding probability.

In the usual normal probability table for $P(z<-2.56)$look at the row that starts with $-2.5$and the column that starts with$.06$

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

Therefore,

Mrs. Starnes solves a simple Sudoku puzzle in under $3$ minutes, or $0.0052$ of the time.

Part (b) Step 1: Calculation

Calculate the $z-$score,

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

To find the equivalent probability, use the normal probability table in the appendix.

The typical normal probability table for $P(z<0.78)$has a row that starts with $0.7$and a column that starts with$.08$

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

Therefore,

Mrs. Starnes spends more than $6$ minutes on an easy task around $21.77$ percent of the time.

Part (c) Step 1: Calculation 

Calculate the $z-$score,

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

Or

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

To find the equivalent probability, use the normal probability table in the appendix.

The usual normal probability table for $P(z<0.78)$has a row that starts with $0.7$and a column that starts with$.08$

Or

In the standard normal probability table for $P(z<3.00)$look for the row that starts with $3.0$and the column that starts with$.00$

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

Therefore,

Mrs. Starnes takes between $6$ and $8$ minutes to finish $21.64$ percent of the easy Sudoku puzzles.