Question 1197972
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Part (a)


At 99% confidence, the z critical value is roughly z = 2.576
Use a table like this
<a href = "https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf">https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf</a>
to get that value. Look at the bottom row labeled "Z" and above the 99% confidence level.


You can also use a stats calculator or spreadsheet to determine this z critical value.


The desired margin of error is 6%, which means we want E = 0.06 or smaller.


We're not told the value of phat, which is the sample proportion of businesses that plan to hire additional employees in the next 60 days. 


Use phat = 0.5 as a conservative estimate. This is the default value of phat if none is stated.


n = min sample size
n = phat*(1-phat)*(z/E)^2
n = 0.5*(1-0.5)*(2.576/0.06)^2
n = 460.817778  approximately
n = 461  always round UP to the nearest whole number


Here's why we round up to the nearest whole number.
Let's try n = 460 in the margin of error formula
E = z*sqrt(phat*(1-phat)/n)
E = 2.576*sqrt(0.5*(1-0.5)/460)
E = 0.060053
We're slightly over the 6% target.
Now try n = 461
E = z*sqrt(phat*(1-phat)/n)
E = 2.576*sqrt(0.5*(1-0.5)/461)
E = 0.059988
Now the error is either 6% or less, which meets the goal we're after.
This is why we round up to the nearest whole number for min sample size problems. This is to clear the hurdle needed.



Answer: <font color=red>461 employers</font>


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Part (b)


Repeat the same set of steps as part (a), but this time we use E = 0.04
Keep everything else the same.


n = phat*(1-phat)*(z/E)^2
n = 0.5*(1-0.5)*(2.576/0.04)^2
n = 1036.84
n = 1037 


Answer: <font color=red>1037 employers</font>


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Part (c)


Now use E = 0.01


n = phat*(1-phat)*(z/E)^2
n = 0.5*(1-0.5)*(2.576/0.01)^2
n = 16589.44 
n = 16590


We need to sample a lot more employers at this point (more than ten times as much compared to the result of part (b)), so it's more practical to go with the 6% or 4% margin of error instead. 


Answer: <font color=red>Minimum sample size gets way too large</font>
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