The confidence interval is arguably one of the most important population measures. According to Reed (2018), the primary role of calculating the confidence interval is to determine the accuracy of a point estimate, for example population means. Different assumptions made during the calculation of confidence intervals influences the procedures used to estimate the metric. One of the most common assumptions made in estimating the confidence interval is that the variance, an important population measure, is unknown. Therefore, it is vital to describe the procedures for estimating a confidence interval when the population variance is unknown.
Step 1: Identify the study population
The first step in calculating a confidence interval is to identify a specific population. The population refers to the entire group of observations that will be studied, for example a collection of weights, age, or blood pressure.
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Step 2: Compute the mean of the population
Mean refers to the average of a population that is calculated by adding the values together and dividing them by the observations.
Formula: Mean (X)= ∑X/n
Step 3: Compute the standard deviation
Since the variance is unknown, it is not possible to know whether the population is normally distributed. According to Reed (2018), data is normally distributed when all the values are closely concentrated around the mean, and they have equal statistical values like mean, median, and mode. The distribution of normal data is shows through bell-shaped curve that is perfectly symmetrical dividing the data into two equal halves ( Steiger, & Fouladi, 2016) . Moreover, normally distributed data is asymptotic, thus the variables can spread to the extreme ends of the bell-shaped curve without touching the axis. Since the variance of the population is unknown, the basic assumption is that it is not normally distributed. Therefore, the standard deviation will be calculated for a non-normal distribution.
The sample variance is calculated using the formula below
S 2 = (X-X) 2 / (N-1)
Where
S 2 = Variance
X=Variable
X= Sample mean
N=Number of observations
Step 4: Determine the test static for calculating the sample’s confidence interval
When the standard deviation is unknown the Z distribution for normal distribution is not applicable; instead the T distribution should be used. Therefore, the focus changes from calculating the population variance to the sample variance ( Krishnamoorthy, 2016) . Therefore, the t-statistic will be used in the calculation
Step 5: Calculate the T critical value using the distribution table that will be used to evaluate the calculated critical value.
The first step in calculating the critical value is subtracting one from the total number of observations in the sample to determine the degree of freedom. For example, if the sample size is 10 the value will be 10-1=9. Step two is to determine the confidence rate that will be used to calculate the alpha using the formula below. The next step is to determine whether the t-test will be one-tail or two tails; if the test is two-tailed them it is important to divide alpha by 2. The last step is reading the t critical value from the table where it is found at the intersection of the degrees of freedom and the alpha value.
α=1-Confidence interval
Step 6: Calculate the standard error
Standard error= t*√S 2 /N
Step 7: Calculate the confidence interval using the formula below
Confidence interval= X + t*√S 2 /N
Where
T= t critical value
X= sample mean
S= Standard deviation
N= sample size
References
Krishnamoorthy, K. (2016). Handbook of statistical distributions with applications . Chapman and Hall/CRC.
Rees, D. G. (2018). Essential statistics . Chapman and Hall/CRC.
Steiger, J. H., & Fouladi, R. T. (2016). Noncentrality interval estimation and the evaluation of statistical models. What if there were no significance tests , 197-229.
