The onesample Ztest is a statistical hypothesis test used to determine whether the mean of a single sample is significantly different from a known population mean. It is used to draw conclusions about a population based on sample data.
In this article, we will discuss the concept of a onesample ztest in R with practical examples.
Why Use the OneSample ZTest
The onesample ztest statistical tool can be used:
 Hypothesis Testing: It helps you to assess whether your sample data provides enough evidence to support or reject a hypothesis about a population parameter.
 Quality Control: ZTests can be used to ensure product quality meets specified standards.
 Research Validation: ZTest can be used to validate research findings and draw a conclusion about the entire population based on sample data.
What is the Formula of OneSample ZTest
The formula for onesample ZTest statistic is:
z < (x̄  μ) / (σ / sqrt(n))
Where:

x̄: observed sample mean

μ: population mean under the null hypothesis

n: sample size

σ: population standard deviation
How to Perform OneSample ZTest in R
To conduct a onesample ztest in R, follow these steps:
Step 1: Collect and Prepare Data
Collect your sample data and ensure it meets the assumption of normality. If the sample size is sufficiently large (typically n > 30), the Central Limit Theorem allows you to assume normality.
Step 2: Formulate Hypotheses
Define your null (Ho) and alternative Hypotheses (Ha):
Null Hypothesis (Ho): The sample mean is equal to the population mean (μ).
Alternative Hypothesis (Ha): The sample mean is not equal to the population mean (μ).
Step 3: Set Significance Level
Choose a significance level (α) typically 0.05, to determine the threshold for statistical significance.
Step 4: Calculate the Test Statistic
In R, you can use the following formula to calculate the Ztest statistic:
z < (x̄  μ) / (σ / sqrt(n))
Step 5: Determine the Critical Value
Find the critical Zvalue using a Ztable or R function like ‘qnorm()‘ based on the chosen significance level (α) and the twotailed nature of the ZTest.
Step 6: Perform the Test
Compare the calculated ZTest statistic in R to the critical value:

If the Z > Critical Value: Reject the null hypothesis.

If the Z ≤ Critical Value: Fail to reject the null hypothesis.
How to Perform OneSample ZTest in R
Let’s understand the onesample ZTest in R with a practical example.
Suppose we have a sample of 50 students, and we want to determine if the average test score differs significantly from the population mean test score of 75.
# Sample data
sample_scores < c(72, 78, 82, 70, 76, 79, 85, 68, 74, 77,
73, 71, 80, 81, 75, 79, 76, 72, 77, 78,
76, 70, 74, 73, 78, 75, 72, 70, 79, 81,
77, 79, 76, 74, 78, 82, 75, 72, 76, 73,
80, 78, 75, 70, 72, 74, 79, 75, 76, 78)
# Perform the onesample Ztest
pop_mean_score < 75
# Choose significance level
alpha < 0.05
# Calcualte the Test statistic
z < (mean(sample_scores)  pop_mean_score) / (sd(sample_scores) / sqrt(length(sample_scores)))
# Calculate critical Zvalue for a twotailed test
critical_value < qnorm(1  alpha / 2)
# Compare the test statistic to the critical value
if (abs(z) > critical_value) {
cat("Reject the null hypothesis: The sample mean is significantly different from the population mean.\n")
} else {
cat("Fail to reject the null hypothesis: The sample mean is not significantly different from the population mean.\n")
}
The output of the above onesample ztest in the R program is:
Fail to reject the null hypothesis: The sample mean is not significantly different from the population mean.
Conclusion
I hope the above article on onesample ztest in R is helpful to you. The onesample ztest is a valuable statistical tool for conducting hypothesis testing, and drawing conclusions about population parameters based on the sample data.