A z-test is a statistical test used to determine whether there is a significant difference between a sample statistic and a population parameter, under the assumption that the population variance is known.
population variance is known: The Z-test is used for hypothesis testing when the population variance is known and the sample size is large (usually n > 30). These conditions allow for a simpler introduction to hypothesis testing because the standard normal distribution (Z-distribution) is used.
Standard Normal Distribution: The Z-test introduces the concept of the standard normal distribution, a critical foundational concept in statistics. Understanding how to standardize a score using the Z-distribution is fundamental and applies to many areas in statistics.
Large Sample Assumption: The Z-test is applicable in scenarios where the sample size is large enough to approximate the sampling distribution of the mean as normally distributed, due to the Central Limit Theorem. This concept is easier for beginners to grasp before moving on to situations where the sample size is small, and the population variance is unknown.
\[Zscore = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}\]
In the formula for the z-test, the variables represent the following components:
\(\bar{x}\) (Sample Mean): This is the mean (average) of the sample data. It represents the average value of the data points in the sample you’re testing.
\(\mu\) (Population Mean): This is the mean (average) of the entire population or the mean of the distribution under the null hypothesis. It’s the value you compare the sample mean against to determine if there is a significant difference.
\(\sigma\) (Population Standard Deviation): This is the standard deviation of the entire population. It measures the dispersion or variability of the population data points around the population mean (\(\mu\)). In the context of a z-test, it’s assumed to be known.
\(n\) (Sample Size): This is the number of observations or data points in the sample. It’s used to calculate the standard error of the mean.
\(z\) (Z-score): The result of the z-test formula. The z-score is a statistical measurement that describes a value’s relationship to the mean of a group of values, measured in terms of standard deviations from the mean. In the context of hypothesis testing, the z-score represents how many standard deviations the sample mean is from the population mean.
The numerator (\(\bar{x} - \mu\)) represents the difference between the sample mean and the population mean. This difference shows how far the sample mean deviates from the population mean, which you’re testing for significance.
The denominator (\(\sigma / \sqrt{n}\)) is the standard error of the mean, which adjusts the population standard deviation (\(\sigma\)) for the size of the sample (\(n\)). This standard error measures how much the sample mean (\(\bar{x}\)) is expected to vary from one sample to another, purely by chance.
Let’s consider a research question related to the average weight of adults in a particular city. Suppose previous studies indicate that the average weight of adults in this city is 75 kgs. We want to test if a new sample of adults from a specific neighborhood has a significantly different average weight, suggesting the neighborhood might have factors influencing weight
Is the average weight of adults in this specific neighborhood different from the general city average of 75 kg?
given that the population standard deviation is known to be 10 kg.
Suppose we collect a sample of 31 adults from the neighborhood and measure their weights. The sample provides the following statistics:
Given the sample values
72, 75, 71, 74, 78, 79, 72, 73, 76, 77, 70, 79, 71, 74, 78, 72, 75, 71, 74, 78, 79, 72, 73, 76, 77, 70, 79, 71, 74, 78, 72Let’s calculate the sample mean, Z-statistic, p-value for the provided data set.
let’s first determine the sample size \(n\) and then calculate the sample mean \(\bar{x}\):
\[ \bar{x} = \frac{\text{Sum of all sample values}}{n} \]
\[\bar{x} = \frac{72 + 75 + 71 + 74 + 78 + 79 + 72 + 73 + 76 + 77 + 70 + 79 + 71 + 74 + 78}{15}\]
The sample mean (\(\bar{x}\)) is 74.52 kg
Given:
We will use the Z-test formula: \[ Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \]
\[ Z = \frac{74.52 - 75}{10 / \sqrt{31}} \approx -0.269 \]
The calculated Z-statistic is approximately -0.269.
To determine if this difference is statistically significant, we would compare the absolute value of the Z-statistic to a critical value from a Z-table, typically at a significance level of 0.05 (for a two-tailed test, this critical value is approximately ±1.96).
Download the Excel file link here
[1] 31# Calculate the sample mean
sample_mean <- mean(sample_weights)
sample_mean [1] 74.51613# Define population parameters
population_mean <- 75
population_sd <- 10
# Calculate the Z-statistic
z_statistic <- (sample_mean - population_mean) / (population_sd / sqrt(sample_size))
z_statistic[1] -0.269408[1] 0.7876158Do not reject null hypothesispython
import numpy as np
from scipy.stats import norm
sample_weights = np.array([72, 75, 71, 74, 78, 79, 72, 73, 76, 77, 70, 79, 71, 74, 78, 72, 75, 71, 74, 78, 79, 72, 73, 76, 77, 70, 79, 71, 74, 78, 72])
alpha = 0.05
# Sample size
sample_size = len(sample_weights)
sample_size31# Sample mean
sample_mean = np.mean(sample_weights)
sample_mean74.51612903225806# Population parameters
population_mean = 75
population_sd = 10
# Z-statistic
z_statistic = (sample_mean - population_mean) / (population_sd / np.sqrt(sample_size))
z_statistic-0.2694079530401626# P-value for the two-tailed test
p_value = 2 * norm.sf(np.abs(z_statistic))
p_value0.7876157667022055if p_value < alpha:
print("Reject null hypothesis")
else:
print("Do not reject null hypothesis")Do not reject null hypothesis.