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Confidence Intervals

What is Confidence Intervals?
The confidence interval is a range of intervals that are likely to include the values from the population with a certain degree of confidence. It often lies between two upper and the lower interval and expressed in the percentage.

How to find confidence intervals?
Step 1: As a first step subtract 1 from the sample size. It is considered as the degree of freedom.
Step 2: Subtract the given confidence level from 1 and divide by 2.
Step 3: Using the values from step 1 and step 2 predict the value from the t-distribution table.
Step 4: Samples standard deviation should be divided by the square root of the sample size.
Step 5:  Multiply the value obtained from step 3 and step 4
Step 6: For the lower end subtract the step 5 value from the mean.
Step 7: For the upper range add the step 5 value to mean.

From the steps, you can understand that the confidence interval depends on the 3 values,

  1. The standard error of measurements
  2. Degree of freedom
  3. The value obtained from the distribution table.

Confidence Intervals and Hypothesis Testing
There is often a good relationship between the confidence interval and the hypothesis testing. If the null hypothesis H0 is rejected in some significance level α then the confidence interval does not contain the value µ0 in the hypothesis at the confidence level of (1-α).

Duality with Confidence Intervals
Below are the examples to demonstrate the relationship between the confidence intervals and the hypothesis testing.

Example 1:
Given: There exists the 95% confidence that the sample mean value lies between 121 and 130.
Null hypothesis: The mean value is 123
Alternate Hypothesis: The mean value is not equal to 123
Since the mean value 123 lies within the confidence interval, the null hypothesis cannot be rejected.

Example 2
For the same above example let us consider the mean value to be 131.In such cases,
Null Hypothesis: The mean value is 131.
Alternate Hypothesis: The mean value is not equal to 131.
The null hypothesis is rejected because the mean value 131 does not lie within the confidence interval.

Relationship between Confidence level and the Significance level for Hypothesis Testing
Confidence Level = 1 – Significance Level.
For example, if the significance level is 0.05 then the confidence level is,
Confidence Level = 1-0.05 = 0.95 = 95%
Now the Confidence level is known and the hypothesis can be tested.
Prediction Based on the P-value
In many cases where the given P-value is less than the α (Significance level), then the confidence interval will not have the value mentioned in the null hypothesis, so the test is significant.

Reference

https://www.mathsisfun.com/data/confidence-interval.html
https://www.austincc.edu/mparker/1342/dl_sp10_pes_sr/dl_09/ch11-17/ht/hyp_conf.htm
https://www.statisticshowto.com/probability-and-statistics/confidence-interval/

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