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Sure, let’s walk through an example of a t-test.

Let’s say we’re working with a software company that claims the average load time for its website is 3 seconds. We think it might be longer, so we collect load times from a sample of 15 different networks. The sample data we collected is as follows:

Our null hypothesis (H0​) is that the average load time is 3 seconds, as the company claims. The alternative hypothesis (Ha​) is that the average load time is greater than 3 seconds.

To conduct a one-sample t-test, we need to calculate the t statistic using the following formula:

where:

• X is the sample mean,
• $\mu$ is the population mean (3 seconds according to the company),
• $s$ is the sample standard deviation,
• $n$ is the sample size (15 in this case).

We’ll use a significance level of 0.05. If our calculated t-value is greater than the critical t-value (for a one-tailed test with $\alpha =0.05$ and 14 degrees of freedom, the critical t-value is approximately 1.761), we’ll reject the null hypothesis.

Let’s perform the calculations.

import numpy as np
from scipy import stats

# given values
load_times = np.array([3.6, 3.9, 4.1, 3.7, 3.2, 3.1, 3.5, 3.8, 3.7, 3.3, 3.4, 3.9, 3.6, 3.2, 3.5])
pop_mean = 3 # population mean

# calculate sample mean and standard deviation
sample_mean = np.mean(load_times)
sample_std = np.std(load_times, ddof=1) # ddof=1 for sample std deviation
n = len(load_times) # sample size

# calculate t-score
t = (sample_mean - pop_mean) / (sample_std / np.sqrt(n))
t, stats.t.sf(t, df=n-1) # sf is survival function, equivalent to 1 - cdf

(7.517205590395159, 1.4023639205416641e-06)