Question 1196501
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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
<a href = "https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf">https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf</a>
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 <= p <= 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 <= E <= 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 = <font color=red>139</font>
With minimum sample size problems, <u>always</u> 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.


<font color=red>Answer: 139</font>
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