Confidence Interval:
A Numerical Example
Measures of central tendencies are nothing but the measures to describe the "central" values of a
collected sample. For an ungrouped set of data these measures are:
the mean,
the median,
and the mode.
You may have to compute the mean of a sample in order to estimate the mean of the population.
If you know the population mean, then it is easy for you to understand how well the sample
mean estimates the population mean.
However, in practice, one would not know the population mean. This is exactly the reason why
one would like to estimate it using a sample drawn from the population.
Obviously our estimate of the population mean will not be perfect. So we need to assign some level
of confidence to our estimate. Let us explain this with an example. Suppose we are assigned a task
of estimating an average science marks scroed by students in year-6.
We assume that we do not know the population mean and variance. So we went ahead and drew a random
sample of 4 students (n). Their marks were 10,23,32,and 34.
The mean of the sample (M) is 24.75 and the standard deviation (S) of the sample is
10.94. As we mentioned in the
cofidence interval description page, we need to calculate the standard error of the
sample mean (SEM). This is given by:
SEM = S/sqrt(n) = 10.94/2 = 5.47
To calculate 95% confidence interval, we need to calculate z-value corresponding to t-distribution
for 95% level and (4-1) degrees of freedom. This z-value is 3.18.
[If the population variance is known, then we need not calculate the sample variance S
to calculate the standard error of the mean (SEM). In this case we need to refer to the standard normal
distribution instead of t-distribution to calculate z-value. For normal distribution 95% of
the values fall between 1.96 times the standard deviation. Since the distribution is standard
normal, the z-value would be just 1.96.]
Since the population variance is unknown in this example, we use z-value of 3.18 from
t-distribution for 95% level and (4-1) degrees of freedom. Therefore the confidence limit
for the population estimate is given by: 3.18*S.E = 17.39. Therefore the lower and limits
are:
The lower limit: M - (z-value)*SEM = 24.75 - 17.39 = 7.36
The upper limit: M + (z-value)*SEM = 24.75 + 17.39 = 42.14
In other words, if we were to repeat this experiment over and over again then in 95% of
the cases the sample mean could be expected to fall within the range of values 7.36
and 42.14
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