Question 1200407
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Answer: <font color=red size=4>0.0314</font>



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Explanation:


On your TI84 calculator, press the button labeled "2nd".
Then press the VARS key.


This brings up the stats distribution menu.
Scroll down to <font color=blue>invNorm</font>


There are two templates for this function.
<font color=blue>invNorm(p, mu, sigma)</font>
and
<font color=blue>invNorm(p)</font>
where
p = area under the normal curve
mu = mean
sigma = standard deviation


The second template is much easier to work with. We only need to worry about one input. 


The second template uses the default values of mu = 0 and sigma = 1, both of which apply to the standard normal Z distribution. 
Luckily we're working with this distribution so we'll use the second template.


A 95% confidence level means we have some positive number k such that
P(-k < Z < k) = 0.95
The area in one tail is (1-C)/2 = (1-0.95)/2 = 0.025


If you were to type <font color=blue>invNorm(0.025)</font> into your TI84 calculator, then you would get approximately: -1.959963986


This means,
P(Z < -1.959963986) = 0.025 approximately
and
P(Z > 1.959963986) = 0.025 approximately
which leads to
P(-1.959963986 < Z < 1.959963986) = 0.95 approximately


Many reference sheets, textbooks, teachers, etc will round this to 2 or 3 decimal places.
However, I'll round it to 5 decimal places which is one more than the "4" mentioned in "Give your answer to four decimal places if possible". That way we have better accuracy.


-1.959963986 rounds to -1.95996 when rounding to 5 decimal places.
We then erase the negative sign to arrive at z = 1.95996 


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z = 1.95996 = critical value
n = 320 = sample size
phat = 0.91 = sample proportion
phat's job is to estimate the population proportion p.


The margin of error formula for a proportion is:
E = z*sqrt(phat*(1-phat)/n)
make sure you use phat and not p.
This is because we don't know what p is.


Another way to render this formula is to say
*[tex \Large E = z*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}]


So,
E = z*sqrt(phat*(1-phat)/n)
E = 1.95996*sqrt(0.91*(1-0.91)/320)
E = 0.03135553171946
E = <font color=red>0.0314</font> which is the final answer.


If you are curious, here is a table of various critical values close to 1.959963986 (rounded to different levels of precision) compared to the error E.
<table border = "1" cellpadding = "5"><tr><td>z</td><td>E before rounding</td><td>E after rounding</td></tr><tr><td>1.96</td><td>0.031356171641321</td><td>0.0314</td></tr><tr><td>1.95996</td><td>0.031355531719451</td><td>0.0314</td></tr><tr><td>1.959964</td><td>0.031355595711638</td><td>0.0314</td></tr><tr><td>1.95996399</td><td>0.031355595551658</td><td>0.0314</td></tr><tr><td>1.959963986</td><td>0.031355595487665</td><td>0.0314</td></tr></table>
When I say "after rounding" I specifically mean "after rounding to 4 decimal places". 
Each item in the far right column is <font color=red>0.0314</font>
I used LibreOffice Calc spreadsheet to help create this table.
That table shows us that we could have gotten away with using z = 1.96, and we would have still arrived at the correct final answer of <font color=red>0.0314</font>


Extra info:
The TI84 calculator can handle rounding. Press the button labeled "MATH" (in my opinion the worst thing to label a button on a MATH calculator), then scroll one unit to the right. 
This lands you in the "NUM" category. Then scroll down to "round("
The template here is <font color=blue>round(x,n)</font> where x is the number you want to round and n is the number of decimal digits.
Type in <font color=blue>round(Ans,4)</font> to round the previous answer to 4 decimal places.
To access the "Ans" variable, press "2nd" then the minus sign button. The button should have a blue "Ans" above it.


Further Reading
<a href = "https://www.statology.org/invnorm-ti-84/">https://www.statology.org/invnorm-ti-84/</a>
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