First, you learnt how you can use your knowledge of the **CLT** to **infer the population mean from the sample mean**.

We estimated the mean commute time of 30,000 employees of an office by taking a sample of 100 employees, finding their mean commute time. Specifically, you were given a sample with a sample mean ¯X = 36.6 minutes and a sample standard deviation S = 10 minutes.

Using the CLT, you concluded that the sampling distribution for the mean commute time would have the following:

- Mean = μ {unknown}

- Standard error = σ√n≈S√n=10√100=1

- Since n(100) > 30, the sampling distribution is a normal distribution.

Using these properties, you were able to claim that the probability that the population mean μ lies between 34.6 (36.6 – 2) and 38.6 (36.6 + 2) is 95.4%.

Then, you learnt the following terminology related to the claim:

- The probability associated with the claim is called the
**confidence level**. (Here, it is 95.4%.).

- The maximum error made in a sample mean is called the
**margin of error**. (Here, it is 2 minutes.).

- The final interval of values is called the
**confidence interval**. [Here, it is the range (34.6, 38.6).].

You then generalised the whole process. Let’s say you have a sample with a **sample size n, mean **and** standard deviation S**. You learnt that a **y% confidence interval** (i.e., a confidence interval corresponding to a y% confidence level) for will be given by the range:

Confidence interval = ,

Where, **Z*** is the Z-score associated with a **y% confidence level**.