Question 1206695
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p = population proportion
phat = sample proportion
The job of phat is to estimate p.


At 95% confidence, the z critical value is roughly z = 1.96
This is something to memorize or have on a reference sheet.


You can use a table such as 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 determine the z critical values. Look at the bottom row and at the value just above the 95% confidence level.


What this means is that P(-1.96 < z < 1.96) = 0.95 approximately.


Another way to determine this z critical value is to use a stats calculator such as a TI84. 


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n = 390 = sample size
x = 12 employees failed
phat = sample proportion of those who failed
phat = x/n
phat = 12/390
phat = 0.03076923 approximately
Around 3.077% of the sample of employees failed the test.
This is the center of the confidence interval.


E = margin of error
E = z*sqrt(phat*(1-phat)/n)
E = 1.96*sqrt(0.03076923*(1-0.03076923)/390)
E = 0.01713940 approximately
This helps determine how wide or spread out the confidence interval is.


L = lower bound of confidence interval
L = phat - E
L = 0.03076923 - 0.01713940
L = 0.01362983
L = 0.014


U = upper bound of confidence interval
U = phat + E
U = 0.03076923 + 0.01713940
U = 0.04790863
U = 0.048


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Answer:
The confidence interval in the format (L, U) would be approximately <font color=red>(0.014, 0.048)</font>


This is equivalent to writing <font color=red>0.014 < p < 0.048</font> which provides more context of which parameter we're trying to estimate. 
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