Confidence Interval: Overview, Questions, Preparation

In Statistics, the confidence interval is a measure by which we can know the exact value of a parameter. Using the confidence level, we can determine the required parameter for the interval depending on the Standard Normal Distribution or the Z-Score.

Thus, confidence levels are acceptable confidence intervals that hold the value of the unknown parameter.

The Formula for Determining the Confidence Interval

We can determine the confidence interval of a dataset using its mean and standard deviation. So, the formula for the confidence interval is:

CI = X̄ ± Zα/2 × [ σ / √n].

Here, X̄ is the Mean of the dataset, Z is the Confidence Coefficient, α is the Confidence Level, σ is the Standard Deviation, and n is the sample space.

Confidence Interval is a part of the chapter ‘Statistics’, prescribed for class 11. The chapter carries 10 marks in the annual examination.

1. If there are 100 apples in a tree and you choose 46 at random, having a mean of 86 and a standard deviation of 6.2, find out whether the apples are big enough. 
Solution:

We know that the Mean = 86 and the Standard Deviation = 6.2. We also know that the number of observations = 46.
If we assume the confidence level to be 95%, then according to the formula to determine the confidence interval we have,
X̄ ± Zα/2 × [ σ / √n].
Or 86 ± 1.96 (From the confidence interval table) * [6.2 / √46]
Hence, we get 86 ± 1.96 * [6.2 / 6.78].
Or 86 ± 1.96 * 0.914
Or, 86 ± 1.79
Therefore, by considering the margin of error, we get the confidence interval to be 84.21 and 87.79.

2. Find the average age of the victims of chain-snatching incidents in a city during a year when the sample size is 100, and the mean age is 34.25 years. 
Solution:
We know that n = 100.
Mean = 34.25 years.
Now, if we assume that the confidence level is 95%, the z-value becomes 1.96.
By substituting these values in the equation for Confidence Interval, we get,
CI = X̄ ± Zα/2 × [ σ / √n].
Therefore, CI = 34.25 ± 1.96 (10 / √100).
Or C.I. = 34.25 ± 1.96.
So, we get the upper limit for the confidence interval to be 34.25 + 1.96 = 36.21.
And the lower limit to be 34.25–1.96 = 32.29.

3. In a survey of 100 Americans and Brits who watched television, the Americans had a standard deviation of 5 while the Brits had a standard deviation of 10. If the sample mean was 35, find out who watched the most TV on average. 
Solution:

CI = X̄ ± Zα/2 × [ σ / √n].
Now, assuming that the confidence level is 95% for the USA, we get,
CI = 35 ± 1.96 * [5 / √100]
Or, CI = 35 ± 1.96 * (5 / 10)
CI = 35 ± 1.96 * 0.5
= 35 ± 0.98 = 35.98 or 34.02. 
Similarly, for the UK: 
CI = 35 ± 1.96 * [10 / √100]
Or CI = 35 ± 1.96 = 36.96 or 33.04.

Q: What is the confidence interval?

Q: What factors affect the confidence interval?

Q: What does a 95% confidence interval mean?

Q: What is the formula for determining the confidence interval for a sample population? 

Q: Why do we use the margin of error?