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:
- CLLower=ˉx−t⋅s√n
- CLUpper=ˉx−t⋅s√n
- where E=t⋅s√n
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:
- t=T.INV.2T(α,df)
- t=T.INV(α2,df)
- E=CONFIDENCE.T(α,2,s,n)
There are two new expressions to understand in above formulas (apart from t):
- df=degrees of freedom=n−1
- α=1−confidence level
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:
α=100%–Confidence Level=1–Confidence Level
We can solve for t using:
t=T.INV.2T(α,df)
Finally, remember that df=n−1
When we use Excel’s T.INV() Function, we input the area to the left of a t-score. The easiest is to input the area in the left tail =α2. This returns the negative (−) t-score:
We can see from the above graph that we can solve for the negative (−) or positive t-scores:
−t=T.INV(α2,df)=T.INV(α2,n−1)
t=T.INV(α2+Conf Level,df)=T.INV(α2+Conf Level,n−1)
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 t-score nor do the margin of error calculation):
E=t⋅s√n=CONFIDENCE.T(α,2,s,n)
We can easily calculate the lower and upper limits once the margin of error is known:
CLLower=ˉx−E
CLUpper=ˉx+E
