Question

Suppose you have four measurement Z=[2,4,6,8], with uncertainty sigma = 1. What is the best estimate value of Z and what is the uncertainty on your best estimate?

Answer 1

Best Estimate ± Uncertainty.
Example: a measurement of 5.07 g ± 0.02 g means that the experimenter is confident that
the actual value for the quantity being measured lies between 5.05 g and 5.09 g.
The uncertainty is the experimenter's best estimate of how far an experimental quantity
might be from the true value.

TRADITIONAL METHOD
given that,
standard deviation, σ =1
sample mean, x =5
population size (n)=4
I.
standard error = sd/ sqrt(n)
where,
sd = population standard deviation
n = population size
standard error = ( 1/ sqrt ( 4) )
= 0.5
II.
margin of error = Z a/2 * (standard error)
where,
Za/2 = Z-table value
level of significance, α = 0.05
from standard normal table, two tailed z α/2 =1.96
since our test is two-tailed
value of z table is 1.96
margin of error = 1.96 * 0.5
= 0.98
III.
CI = x ± margin of error
confidence interval = [ 5 ± 0.98 ]
= [ 4.02,5.98 ]
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DIRECT METHOD
given that,
standard deviation, σ =1
sample mean, x =5
population size (n)=4
level of significance, α = 0.05
from standard normal table, two tailed z α/2 =1.96
since our test is two-tailed
value of z table is 1.96
we use CI = x ± Z a/2 * (sd/ Sqrt(n))
where,
x = mean
sd = standard deviation
a = 1 - (confidence level/100)
Za/2 = Z-table value
CI = confidence interval
confidence interval = [ 5 ± Z a/2 ( 1/ Sqrt ( 4) ) ]
= [ 5 - 1.96 * (0.5) , 5 + 1.96 * (0.5) ]
= [ 4.02,5.98 ]
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interpretations:
1. we are 95% sure that the interval [4.02 , 5.98 ] contains the true population mean
2. if a large number of samples are collected, and a confidence interval is created
for each sample, 95% of these intervals will contains the true population mean