SOLUTION: If n=440 and X = 352, construct a 99% confidence interval. Give your answers to three decimals ______< p <_______

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Question 1200393: If n=440 and X = 352, construct a 99% confidence interval.
Give your answers to three decimals
______< p <_______

Found 2 solutions by amarjeeth123, math_tutor2020:
Answer by amarjeeth123(569)   (Show Source): You can put this solution on YOUR website!
Mean confidence interval: [-43.167 , 43.167].
Alternatively: 0 ± 43.167
Margin of Error (MOE): 43.167.
Standard Error (S.E): 16.686.
Since you use the sample standard deviation (S), the formula uses the t-distribution with n-1 degrees of freedom.
If you would calculate the confidence interval over an infinite number of samples with a sample size of 440, 99% (CL) of the calculated confidence intervals will contain the mean's true value, and 1% (α) of the calculated confidence intervals will not contain the mean's true value.
Calculation steps
The mean confidence interval formula is:
x̄ ± MOE
x̄ ± T1-α/2(df) * S
√n
Calculate the degrees of freedom:
df = n - 1 = 440 - 1 = 439
Calculate the significance level:
α = 1 - CL = 1 - 0.99 = 0.01.
Calculate the probability (p):
p= 1 - α/2 = 1 - 0.01/2 = 0.995.
If we don't know the population standard deviation we have to use the t-distribution only.
Calculate the t-score:
T0.995(439) = 2.587
x̄ ± T0.995(439) * 350
√440
0 ± 2.587 * 350
√440
0 ± 2.587 * 16.686
0 ± 43.167
Since Tα/2 = -T1-α/2 , you may use Tα/2 instead of T1-α/2
Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

The other tutor has made a few mistakes. Here is the proper way to solve this problem.
phat = sample proportion
phat = (number of successes)/(sample size)
phat = x/n
phat = 352/440
phat = 0.8 exactly
This is the center of the confidence interval.
The job of phat is to estimate the population proportion p.

At 99% confidence, the z critical value is roughly z = 2.576
This value is to either be memorized or looked up on a reference sheet.
The back of your stats textbook will have the necessary table of values.
Your stats professor may let you have a notecard for exams, or perhaps hand out reference sheets for exams.

E = margin of error for a proportion
E = z*sqrt(phat*(1-phat)/n)
E = 2.576*sqrt(0.8*(1-0.8)/440)
E = 0.049122392598 approximately
E = 0.049122

L = lower boundary of confidence interval
L = phat - E
L = 0.8 - 0.049122
L = 0.750878
L = 0.751

U = upper boundary of confidence interval
U = phat + E
U = 0.8 + 0.049122
U = 0.849122
U = 0.849

The values of L and U are approximate.

The 99% confidence interval of the format L < p < U is therefore approximately 0.751 < p < 0.849 when rounding to 3 decimal places.
Many stats calculators can be used to confirm this solution.
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