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Sure, let’s walk through an example of a Z-test.

Imagine we’re working for a company that manufactures light bulbs. The company claims that its light bulbs have an average lifespan of 1,500 hours. We suspect this might not be accurate, so we conduct a test. We take a sample of 50 light bulbs and find that the average lifespan is 1,480 hours with a standard deviation of 100 hours.

Our null hypothesis, H0​, is that the average lifespan of the light bulbs is 1,500 hours, as the company claims. The alternative hypothesis, Ha​, is that the average lifespan is not 1,500 hours.

Let’s conduct a Z-test to test our hypothesis. The Z-statistic is calculated as follows:

​

where:

• X is the sample mean (1,480 hours in our case),
• $\mu$ is the population mean (1,500 hours according to the company),
• $\sigma$ is the standard deviation of the sample (100 hours),
• $n$ is the sample size (50 light bulbs).

We’ll use a significance level of 0.05. If our calculated Z-score is beyond the critical Z-score (approximately ±1.96 for a two-tailed test with $\alpha =0.05$), we’ll reject the null hypothesis.

Let’s calculate.

import math

# given values
sample_mean = 1480 # sample mean
pop_mean = 1500 # population mean
std_dev = 100 # standard deviation
n = 50 # sample size

# calculate z-score
z = (sample_mean - pop_mean) / (std_dev / math.sqrt(n))
z