Use the given data to find the minimum sample size required to estimate the population proportion.
margin of error: 0.008; confidence level: 98%; modifyingabove p with caret and modifyingabove q with caret unknown

The minimum sample size required is 21134.

Step-by-step explanation:

Consider the provided information.

It is given that the Margin of error: 0.008; confidence level: 98%

Thus, M.E = 0.008,

are unknown. Let us assume .

In order to find the value of "n" use the formula:

For the value of  

Substitute the respective values in the above formula.

Hence, the minimum sample size required is 21134.

37.5 liters of 32% solution

Step-by-step explanation:

Assign variables

let "a" be the number of liters to add from the 32% alcohol solution

let "c" be the number of liters to have a 46% alcohol solution

Create equations using the information given in the problem.

Equation for total liters

a + 15 = c

Equation for percentage

0.32a + 0.81(15) = 0.46c

0.32a + 12.15 = 0.46c

Substitute the expression for total liters into the equation for percentage. Both equations have "c".

0.32a + 12.15 = 0.46c                   Equation for percentage

0.32a + 12.15 = 0.46(a + 15)                 Substituted expression

0.32a + 12.15 = 0.46a + 6.9

Start isolating "a"

0.32a + 12.15 - 12.15 = 0.46a + 6.9 - 12.15     Subtract 12.15 on both sides

0.32a = 0.46a - 5.25

0.32a - 0.46a = 0.46a - 0.46a - 5.25             Subtract 0.46a on both sides

-0.14a = -5.25

-0.14a/-0.14 = -5.25/-0.14                    Divide both sides by -0.14

a = 37.5                      Liters to add from 32% alcohol solution

If you need to know how many liters your final solution is:

Substitute a = 37.5 into the equation for total liters

a + 15 = c  

37.5 + 15 = c                     Add

c = 52.5                 Liters of final solution

Check your answer using the equation for percentage

0.32a + 0.81(15) = 0.46c             Substitute a=37.5 and c=52.5

0.32(37.5) + 0.81(15) = 0.46(52.5)              Multiply each term

12 + 12.15 = 24.15               Add to simplify

24.15 = 24.15            Same answer

LS = RS                   Left side equals right side

Therefore 37.5 liters of the 32% alcohol solution must be added with 15 liters of an 81% alcohol solution.