Confidence Interval for Population Mean (One-Sample t-test)

tags: #statistics/inferential/ttest/one_sample

Formula: CI for one-sample t-test

To compute the CI for a one-sample t-test:

C I (\bar{y}) = \bar{y} \pm t_{n - 1}^{*} \cdot S E (\bar{y}), where S E (\bar{y}) = \frac{s}{\sqrt{n}}

where, $t^{*}$ is the critical value from student’s distribution with $n - 1$ degrees of freedom based on the confidence level.

Computing the CI in Python

We can use the t.interval() function from the scipy.stats library to calculate a confidence interval for a population mean:

from scipy.stats import t

# compute ci interval
t.interval(alpha, df, loc, scale)

Note that the t.interval() function takes the following arguments:

alpha: The desired confidence level, represented as a value between 0 and 1.
df: The degrees of freedom, equal to the sample size minus one.
loc: The sample mean.
scale: The standard error of the sample mean, computed as the sample standard deviation divided by the square root of the sample size.

Example

import numpy as np
from scipy.stats import t

# Generate sample data
sample = np.array([4.3, 3.8, 5.1, 4.9, 3.5, 4.2, 4.6, 4.0, 5.2, 4.7])

# Compute the sample mean and sample standard deviation
n = len(sample)
sample_mean = np.mean(sample)
sample_std = np.std(sample, ddof=1)

# Set the desired confidence level and degrees of freedom
alpha = 0.05
df = n - 1

# Compute the t-value for the desired confidence level and degrees of freedom
t_value = t.ppf(1 - alpha / 2, df)

# Compute the lower and upper bounds of the confidence interval
lower_bound, upper_bound = t.interval(alpha, df, loc=sample_mean, scale=sample_std / np.sqrt(n))

# Print the results
print("Sample mean: {:.2f}".format(sample_mean))
print("Sample standard deviation: {:.2f}".format(sample_std))
print("95% confidence interval: [{:.2f}, {:.2f}]".format(lower_bound, upper_bound))