Q. 70 Toothpaste Ken is traveling for ... [FREE SOLUTION]

91影视

The Practice of Statistics for AP Examination

Daren Starnes, Josh Tabor

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 837 pages

ISBN: 9781319113339

Chapter 6: Q. 70 (page 395)

Toothpaste Ken is traveling for his business. He has a new $0.85$ -ounce tube of toothpaste that鈥檚 supposed to last him the whole trip. The amount of toothpaste Ken squeezes out of the tube each time he brushes is independent, and can be modeled by a Normal distribution with mean $0.13$ ounce and standard deviation $0.02$ ounce. If Ken brushes his teeth six times on a randomly selected trip, what鈥檚 the probability that he鈥檒l use all the toothpaste in the tube?

Short Answer

Expert verified

Ken's chances of finishing the tube of toothpaste are $0.0764$ .

Step by step solution

Given information

The value of the mean

$渭_{X} = 0.13 ounce$

The value of standard deviation,

$蟽_{X} = 0.02 ounce$

The value of Tube weighs,

$x = 0.85 - ounce$

Calculation

If X and Y are independent,

The Property mean:

$渭_{a X + b Y} = a 渭_{X} + b 渭_{Y}$

The Property variance:

$蟽_{a X + b Y}^{2} = a^{2} 渭_{X}^{2} + b^{2} 渭_{Y}^{2}$

Now,

Brushing six times' mean is the sum of brushing six times' mean:

$\begin{array}{rcl} 渭_{X_{1} + X_{2} + X_{3} + X_{4} + X_{5} + X_{6}} & = & 渭_{X_{1}} + 渭_{X_{2}} + 渭_{X_{3}} + 渭_{X_{4}} + + 渭_{X_{3}} + 渭_{X_{6}} \\ = & 6 渭_{X} \\ = & 6 (0.13) \\ = & 0.78 ounce \end{array}$

We know that

The square root of variance is standard deviation:

$\begin{array}{rcl} 蟽_{X_{1} + X_{2} + X_{3} + X_{4} + X_{5} + X_{6}} & = & \sqrt{蟽_{X_{1}}^{2} + 蟽_{X_{2}}^{2} + 蟽_{X_{3}}^{2} + 蟽_{X_{4}}^{2} + 蟽_{X_{5}}^{2} + 蟽_{X_{6}}^{2}} \\ = & \sqrt{6 (0.02)^{2}} \\ 鈮� & 0.0490 ounce \end{array}$