1810DMTCYFS03229 HARSH RIMZA Confidence Interval is a range where we are certain that true value exists. The selection of a confidence level for an interval determines the probability that the confidence interval will contain the true parameter value. This range of values is generally used to deal with population-based data, extracting specific, valuable information with a certain amount of confidence. Probability of including the true value of a parameter within a confidence interval. The confidence level describes the uncertainty associated with a sampling method. For Example : let’s suppose you were surveying an average height of men in a particular city. To find that, you set a 95% confidence level and find that the 95% confidence interval is (168,182). That means if you repeated this over and over, 95 percent of the time the height of a man would fall somewhere between 168 cm and 182 cm. Two extreme measurements within which an observation lies. Or in other words end points of the confidence interval. Step 1: Identify the sample problem. Choose the statistic (like sample mean, etc) that you will use to estimate population parameter. Step 2: Select a confidence level. (Usually, it is 90%, 95% or 99%) Step 3: Find the margin of error. (Usually given) If not given, use the following formula:Margin of error = Critical value * Standard deviation Step 4: Specify the confidence interval. The uncertainty is denoted by the confidence level. And the range of the confidence interval is defined by Eq-1. Confidence Interval = Sample_Statistic + Margin_of_Error where , Sample_Statistic : Can be sample mean. Margin_of_Error : Is generally ± 2.5 -Eq-1 Mean (μ) — Arithmetic mean is the average of numbers. It is defined as the sum of n numbers divided by the count of numbers till n. (Eq-2) μ = (1+2+3+..+n)/n - Eq 2 Standard deviation (σ) — It is defined as the summation of squared of the difference between each number and the mean. (Eq-3) Using t-distribution : We use t-distribution when the sample size n<30. Step 1: Subtract 1 from your sample size.[Eq-4] This gives the degrees of freedom (df) df=n-1 Eq-4 Step 2 - Subtract the confidence interval from 1, then divide by two. [Eq-5] This gives the significance level (α) α=(1- CL)/2 Eq-5 Step 3 - Use the values of α and df in the t-distribution table and find the value of t. Step 4 - Use the t-value obtained in step 3 in the formula given for Confidence Interval with t-distribution. [Eq-6] μ ± t(σ/√n) where, μ = mean, t = chosen t-value from the table above, σ = the standard deviation, n = number of observations. Eq-6 Using z-distribution : We use z-distribution when the sample size n>30. Z-test is more useful when the standard deviation is known. Step 1 : Find the mean. Step 2 : Find the standard deviation. Step 3 : Determine the z-value for the specified confidence interval. Step 4 - Use the z-value obtained in step 3 in the formula given for Confidence Interval with z-distribution. [Eq-7] μ ± z (σ/√n) Eq-7 where, μ = mean, z = chosen z-value from the table above σ = the standard deviation n = number of observations Confidence Interval is one of the foundational concepts of statistics. It tells a statement about the data. Various sampling methods such as mean, median etc. can be used based on the data present. One can also determine what distribution to use when in order to get the best results.