SOLUTION: The brand manager for a brand of toothpaste must plan a campaign designed to increase brand recognition. He wants to first determine the percentage of adults who have heard of the

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Question 1196501: The brand manager for a brand of toothpaste must plan a campaign designed to increase brand recognition. He wants to first determine the percentage of adults who have heard of the brand. How many adults must he survey in order to be 90​% confident that his estimate is within seven percentage points of the true population​ percentage? Complete parts​ (a) through​ (c) below.
​a) Assume that nothing is known about the percentage of adults who have heard of the brand.

Answer by math_tutor2020(3817) About Me  (Show Source):
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At 90% confidence, the z critical value is about 1.645
Use a table like this to determine the critical value
https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf
Look blue row at the bottom marked "Z"
The value 1.645 is just above the 90% confidence level

Since we're assuming we know nothing about the population percentage, this means p = 0.50 is the most conservative estimate.
It's right in the middle between p = 0 and p = 1
The true value of p is somewhere in the interval 0+%3C=+p+%3C=+1, so why not go for the exact middle.

We want the error to be E = 0.07 since he wants to be within 7 percentage points of the true value of p.
This means we want E = 0.07 or smaller
We also can't have negative E values either.
0+%3C=+E+%3C=+0.07

Here's a recap of the values we'll be using
z = 1.645
p = 0.50
E = 0.07

They lead to...
n = p*(1-p)*(z/E)^2
n = 0.50*(1-0.50)*(1.645/0.07)^2
n = 138.0625
n = 139
With minimum sample size problems, always round up to the nearest whole number.
It doesn't matter that 138.0625 is closer to 138 than it is to 139.

Let's compute the margin of error for a sample size of n = 138
E = z*sqrt(p*(1-p)/n)
E = 1.645*sqrt(0.50*(1-0.50)/138)
E = 0.0700158496549
We're slightly over the target of 0.07

Now try n = 139
E = z*sqrt(p*(1-p)/n)
E = 1.645*sqrt(0.50*(1-0.50)/139)
E = 0.06976353946618
Now we're under 0.07
This shows why we rounded up to 139.

Answer: 139