Chapter 6.2, Problem 70E

To determine

Probability for Ken will use all the toothpaste in the tube.

Answer to Problem 70E

Probability that Ken will use all the toothpaste in the tube is 0.0764.

Explanation of Solution

Given information:

Mean,

  ${\mu }_X=0.13\text{ounce}$

Standard deviation,

  ${\sigma }_X=0.02\text{ounce}$

Tube weighs,

  $x=0.85-\text{ounce}$

Calculations:

If X and Y are independent,

Property mean:

  ${\mu }_{aX+bY}=a{\mu }_X+b{\mu }_Y$

Property variance:

  ${\sigma }_{aX+bY}^2=a^2{\mu }_X^2+b^2{\mu }_Y^2$

Now,

Mean of brushing six times is the sum of mean of brushing six times each:

  ${\mu }_{X_1+X_2+X_3+X_4+X_5+X_6}={\mu }_{X_1}+{\mu }_{X_2}+{\mu }_{X_3}+{\mu }_{X_4}++{\mu }_{X_5}+{\mu }_{X_6}=6{\mu }_X=6\left(0.13\right)=0.78\text{ounce}$

We know that

Standard deviation is the square root of Variance:

  ${\sigma }_{X_1+X_2+X_3+X_4+X_5+X_6}=\sqrt{{\sigma }_{X_1}^2+{\sigma }_{X_2}^2+{\sigma }_{X_3}^2+{\sigma }_{X_4}^2+{\sigma }_{X_5}^2+{\sigma }_{X_6}^2}=\sqrt{6{\left(0.02\right)}^2}\approx 0.0490\text{ounce}$

Calculate the z − score:

  $z=\frac{x-\mu }{\sigma }=\frac{0.85-0.78}{0.0490}\approx 1.43$

Use Table − A to find the corresponding probability:

  $\begin{array}{l}P\left(X>0.85\right)=P\left(z>1.43\right)\\ =1-P\left(z<1.43\right)\\ =1-0.9236\\ =0.0764\end{array}$

Thus,

Probability that Ken will use all the toothpaste in the tube is 0.0764.