In this article, we will be looking at the different ways to calculate confidence intervals using various distributions in the Python programming language. Confidence interval for a mean is a range of values that is likely to contain a population mean with a certain level of confidence.
Formula:
Confidence Interval = x(+/-)t*(s/√n)
- x: sample mean
- t: t-value that corresponds to the confidence level
- s: sample standard deviation
- n: sample size
Method 1: Calculate confidence Intervals using the t Distribution
This approach is used to calculate confidence Intervals for the small dataset where the n<=30 and for this, the user needs to call the t.interval() function from the scipy.stats library to get the confidence interval for a population means of the given dataset in python.
Syntax: st.t.interval(alpha, length, loc, scale))
Parameters:
- alpha: Probability that an RV will be drawn from the returned range.
- length: Length of the data set
- loc: location parameter
- scale: scale parameter
Example 1:
In this example, we will be using the data set of size(n=20) and will be calculating the 90% confidence Intervals using the t Distribution using the t.interval() function and passing the alpha parameter to 0.90 in the python.
Python
import numpy as npimport scipy.stats as st # define sample datagfg_data = [1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 6, 7, 8, 10] # create 90% confidence intervalst.t.interval(alpha=0.90, df=len(gfg_data)-1, loc=np.mean(gfg_data), scale=st.sem(gfg_data)) |
Output:
(2.962098014195961, 4.837901985804038)
Example 2:
In this example, we will be using the data set of size(n=20) and will be calculating the 90% confidence Intervals using the t Distribution using the t.interval() function and passing the alpha parameter to 0.99 in the python.
Python
import numpy as npimport scipy.stats as st # define sample datagfg_data = [1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 6, 7, 8, 10] # create 99% confidence intervalst.t.interval(alpha=0.99, df=len(gfg_data)-1, loc=np.mean(gfg_data), scale=st.sem(gfg_data)) |
Output:
(2.3481954013214263, 5.4518045986785735)
Interpretation from example 1 and example 2:
In the case of example 1, the calculated confident mean interval of the population with 90% is (2.96-4.83), and in example 2 when calculated the confident mean interval of the population with 99% is (2.34-5.45), it can be interpreted that the example 2 confident interval is wider than the example 1 confident interval with the 95% of the population, which means that there are 99% chances the confidence interval of [2.34, 5.45] contains the true population mean
Method 2: Calculate confidence Intervals using the Normal Distribution
This approach is used to calculate confidence Intervals for the large dataset where the n>30 and for this, the user needs to call the norm.interval() function from the scipy.stats library to get the confidence interval for a population means of the given dataset where the dataset is normally distributed in python.
Syntax: st.norm.interval(alpha, loc, scale))
Parameters:
- alpha: Probability that an RV will be drawn from the returned range.
- loc: location parameter
- scale: scale parameter
Example 3:
In this example, we will be using the random data set of size(n=100) and will be calculating the 90% confidence Intervals using the norm Distribution using the norm.interval() function and passing the alpha parameter to 0.90 in the python.
Python
import numpy as npimport scipy.stats as st # define sample datagfg_data = np.random.randint(5, 10, 100) # create 90% confidence interval# for population mean weightst.norm.interval(alpha=0.90, loc=np.mean(gfg_data), scale=st.sem(gfg_data)) |
Output:
(6.920661262464349, 7.3593387375356505)
Example 4:
In this example, we will be using the random data set of size(n=100) and will be calculating the 99% confidence Intervals using the norm Distribution using the norm.interval() function and passing the alpha parameter to 0.99 in the python.
Python
import numpy as npimport scipy.stats as st # define sample datagfg_data = np.random.randint(5, 10, 100) # create 99% confidence interval# for population mean weightst.norm.interval(alpha=0.99, loc=np.mean(gfg_data), scale=st.sem(gfg_data)) |
Output:
(6.689075889330163, 7.450924110669837)
Interpretation from example 3 and example 4:
In the case of example 3, the calculated confident mean interval of the population with 90% is (6.92-7.35), and in example 4 when calculated the confident mean interval of the population with 99% is (6.68-7.45), it can be interpreted that the example 4 confident interval is wider than the example 3 confident interval with the 95% of the population, which means that there are 99% chances the confidence interval of [6.68, 7.45] contains the true population means.
