Confidence Interval & Confidence Level
I) IDENTIFYING CONFIDENCE LEVEL GIVEN CONFIDENCE INTERVAL
To find the confidence interval in a given range, just understand that the range given is the z score, and from the z score you can directly find the probability in excel as =NORM.S.DIST(zscore, TRUE). Using this formula find the probability for z score and -zscore. And then subtract them to get the cumulative probability in the range of the errors! You do not need to know the mean of either the sample distribution or the original population! Some common standard confidence intervals are as follows —
Example Question :
Given a sample data where x_bar = 36.6, s = 10, n = 100, what is the confidence interval associated with the claim that the average commute time of the employees is between 34.4–2.58 and 36.6 + 2.58.
Solution :
P(z = + 2.58) = NORM.S.DIST(2.58, TRUE) = 0.99
p(z = -2.58) = NORM.S.DIST(-2.58, TRUE) = 0.004
0.99–0.004 = 0.99 = CONFIDENCE LEVEL
The same thing can be arrived at mathematically as follows —
II) IDENTIFYING CONFIDENCE INTERVAL GIVEN CONFIDENCE LEVEL
Example Question:
n = 100, xbar = 2.3, sd = 0.3. What is the confidence interval for a 99% confidence level.
Solution:
For 99% confidence level if you find NORM.S.INV(0.99) you will not get the requisite 2.58. Rather you will get 2.32. This is because of the decimal numbers round off error. A z-score of 2.58 cannot be retrieved only with NORM.S.INV(0.99506). Because the inverse gives the probability on the left. So, it is better to remember the table above to know the correlation of the confidence level and z scrore.
- Identify the zscore from the mapping of confidence interval and zscore. So, for 99% accuracy, the z score is 2.58
- Now, you need to map this zscore to the actual value in the population scale.
Zscore is actually (random variable value / Standard deviation) for a normal distribution. To be able to find the confidence interval on the actual population, note that the random variable is the actual population and you need to find the Standard deviation of the actual population. Also you need to ensure that the actual population is also a normal distribution!! But that might not be the case. Hence, we need to find another way!
We somehow need to translate from the sampling distribution to the actual population. The bridge to that is Standard Error of the means SEM (sample distribution’s standard deviation).
- SEM = meu/square root of (n) [meu is the actual population mean. That can be approximated to xbar — the mean of the sample distribution as per CLT]
- So, the interval is z-score * SEM. Hence the formula —
