Question 1200407: In a recent poll, 320 people were asked if they liked dogs, and 91% said they did. Find the Margin of Error for this poll, at the 95% confidence level. Give your answer to four decimal places if possible.
*I need help understanding how to solve this using my TI-84 calculator
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
Answer: 0.0314
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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 invNorm
There are two templates for this function.
invNorm(p, mu, sigma)
and
invNorm(p)
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 invNorm(0.025) 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
}{n}})
So,
E = z*sqrt(phat*(1-phat)/n)
E = 1.95996*sqrt(0.91*(1-0.91)/320)
E = 0.03135553171946
E = 0.0314 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.
| z | E before rounding | E after rounding | | 1.96 | 0.031356171641321 | 0.0314 | | 1.95996 | 0.031355531719451 | 0.0314 | | 1.959964 | 0.031355595711638 | 0.0314 | | 1.95996399 | 0.031355595551658 | 0.0314 | | 1.959963986 | 0.031355595487665 | 0.0314 |
When I say "after rounding" I specifically mean "after rounding to 4 decimal places".
Each item in the far right column is 0.0314
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 0.0314
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 round(x,n) where x is the number you want to round and n is the number of decimal digits.
Type in round(Ans,4) 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
https://www.statology.org/invnorm-ti-84/
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