In: Math
find the sample size needed to give a margin of error to estimate a proportion within plus minus 2% within 99% confidence within 95% confidence within 90% confidence assume no prior knowledge about the population proportion p
Solution:
Given:
E = Margin of Error = 2% = 0.02
p is unknown , thus p = 0.5
Part a) 99% confidence level
Formula:
Zc is z critical value for c = 0.99 confidence level.
Find Area = ( 1+c)/2 = ( 1 + 0.99 ) / 2 = 1.99 /2 = 0.9950
Thus look in z table for Area = 0.9950 or its closest area and find corresponding z critical value.
From above table we can see area 0.9950 is in between 0.9949 and 0.9951 and both are at same distance from 0.9950, Hence corresponding z values are 2.57 and 2.58
Thus average of both z values is 2.575
Thus Zc = 2.575
Thus
( Sample size is always rounded up)
Part b) 95% confidence level
Formula:
We need to find zc value for c=95% confidence level.
Find Area = ( 1 + c ) / 2 = ( 1 + 0.95) /2 = 1.95 / 2 = 0.9750
Look in z table for Area = 0.9750 or its closest area and find z value.
Area = 0.9750 corresponds to 1.9 and 0.06 , thus z critical value = 1.96
That is : Zc = 1.96
Thus
Part c) 90% confidence level
Formula:
Zc is z critical value for c = 90% confidence level.
Find Area = ( 1 + c ) / 2 = ( 1 + 0.90) / 2 = 1.90 / 2 = 0.9500
Look in z table for Area = 0.9500 or its closest area and find corresponding z value.
Area 0.9500 is in between 0.9495 and 0.9505 and both the area are at same distance from 0.9500
Thus we look for both area and find both z values
Thus Area 0.9495 corresponds to 1.64 and 0.9505 corresponds to 1.65
Thus average of both z values is : ( 1.64+1.65) / 2 = 1.645
Thus Zc = 1.645
Thus