Calculation of the Confidence Interval
Confidence interval in statistics refers to an estimated interval calculated that a particular parameter value lies between the range. Confidence interval consists of an upper bound and a lower bound. An upper bound is the highest value that a parameter cannot exceed while the lower bound is the one believed that the parameter could not be less than it (Tripathi, 2016). The formula for calculating the confidence intervals is given as Where X is the mean of observations, Z is the Z score value at 95% s is the standard deviation, and n is the total number of observations. We are using a Z score since we have been given the value standard deviation is known. In our problem, we have the following essential data to calculate the confidence interval.
N=6
S=1.615
X=4.5
Z=1.96 at 95% confidence level.
Now
Therefore, an estimated parameter from the observation of mean 4.5 and standard deviation 1.615 has a very high probability of falling between the interval [3.2077, 5.7923] with only 5% benefit of the doubt.
The confidence interval can also be calculated using the method of student t-distribution with given degrees of freedom. However, it cannot be used in this problem since we have been given the standard deviation. T score for the confidence interval is only used when the standard deviation is unknown, and a standard error is used as its estimator.