Confidence Intervals
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Key Questions

Answer:
See explanation.
Explanation:
An estimate of a population parameter may be expressed in two ways:
POINT ESTIMATION
A point estimate of a population parameter is a single value of a statistic.
Example,
The sample mean
#barx# is a point estimate of the population mean Î¼. Similarly, the sample proportion p is a point estimate of the population proportion P.INTERVAL ESTIMATION
An interval estimate is defined by two numbers, between which a population parameter is said to lie.
Example
#a < x < b# is an interval estimate of the population mean#mu# . It indicates that the population mean is greater than a but less than#b# .In any estimation problem, we need to obtain both a point estimate and an interval estimate. The point estimate is our best guess of the true value of the parameter, while the interval estimate gives a measure of the accuracy of that point estimate by providing an interval that contains plausible values.

There are total 3 cases
if sigma is known(doesn't matter n is large or small) then use this formula
#barx z_"(alpha/2)" *sigma/sqrt(n)# for lower confidence interval limit
#barx+ z_"(alpha/2)" *sigma/sqrt(n)# for upper confidence interval limitif n is >30 and sigma is unknown then use this formula
#barx z_"(alpha/2)" *S/sqrt(n)# for lower confidence interval limit
#barx+z_"(alpha/2)" *S/sqrt(n)# for upper confidence interval limitif n is <=30 sigma is unknown then use this formula
#barx t_"(alpha/2,df=n1)" *s/sqrt(n)# for lower confidence interval limit
#barx+ t_"(alpha/2,df=n1)" *s/sqrt(n)# for upper confidence interval limitnow consider the example for large sample size
n=3,500
xbar=174.4
S=38.7
The Z value for 95% confidence is Z=1.96
#174.41.96*38.7/sqrt(3500)#
#174.4+1.96*38.7/sqrt(3500)#
hence
#173.11<=mu<=175.68# now consider the example for small sample size
n=10
xbar=176
s=33
the degrees of freedom (df) = n1 = 9. The t value for 95% confidence with df = 9 is t = 2.262.
#1762.262*33/sqrt(10)#
#176+2.262*33/sqrt(10)#
hence
#152.39<=mu<=190.60# 
Suppose the confidence interval is 95%.
This interval then represents the range within which you would expect a sample mean to fall in 95% of random samples.
It does not mean that the probability is 95%, or that a specific value will fall within this interval in 95% of cases.
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