Confidence Intervals
Confidence Intervals when is σ Unknown
Learning Objectives
In this section, we will do the following calculations to estimate the true mean when the population standard deviation (σ) is unknown:
- Understand when z-scores and when t-scores are used
- Understand the meaning of t-scores
- Examine several Excel calls to calculate t-scores and the margin of error
- Introduce the confidence intervals formula
When the population standard deviation (σ) is unknown, we need to use a t-score instead of a z-score. In this case, the confidence interval formulas become:
- where
Calculating T-scores
When the population standard deviation is unknown, we must use a t-score instead of a z-score (that we used previously). We will explore what t-scores and what a t-distribution is in the next section also. For now, we will examine the 3 possible ways of calculating a t-score or the margin of error related to the t-score in Excel:
There are two new expressions to understand in above formulas (apart from t):
When we use Excel’s T.INV.2T() function, we input the area outside of the confidence interval. This area is also called α (alpha). See figure 49.1 below to better understand this area.
We can see from the above graph that α (alpha) makes up the area in the upper and lower tails (split between the two tails). It can be calculated using:
We can solve for using:
Finally, remember that
When we use Excel’s T.INV() Function, we input the area to the left of a -score. The easiest is to input the area in the left tail . This returns the negative (−) -score:
We can see from the above graph that we can solve for the negative (−) or positive -scores:
When we use Excel’s CONFIDENCE.T() function, we input the area outside of the confidence interval (α). This function returns the Margin of Error (E):
Excel’s CONFIDENCE.T() function is the quickest way to calculate the margin of error (there is no need to calculate the -score nor do the margin of error calculation):
We can easily calculate the lower and upper limits once the margin of error is known:
