## What is One-Sample Z-Test

**The one-sample Z-test 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 one-sample z-test in R with practical examples.

## Why Use the One-Sample Z-Test

The one-sample z-test 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:**Z-Tests can be used to ensure product quality meets specified standards.**Research Validation:**Z-Test can be used to validate research findings and draw a conclusion about the entire population based on sample data.

## What is the Formula of One-Sample Z-Test

The formula for one-sample Z-Test 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 One-Sample Z-Test in R

To conduct a one-sample z-test 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 Z-test statistic:

`z <- (x̄ - μ) / (σ / sqrt(n))`

**Step 5: Determine the Critical Value**

Find the critical Z-value using a Z-table or R function like ‘**qnorm()**‘ based on the chosen significance level (α) and the two-tailed nature of the Z-Test.

**Step 6: Perform the Test**

Compare the calculated Z-Test 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.

## Example 1: Perform One-Sample Z-Test in R

Let’s understand the one-sample Z-Test 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 one-sample Z-test
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 Z-value for a two-tailed 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 one-sample z-test in the R program is:

`Fail to reject the null hypothesis: The sample mean is not significantly different from the population mean.`

**Cool Tip:** How to use a one-sample proportion z-test for categorical data!

## Conclusion

I hope the above article on one-sample z-test in R is helpful to you. The one-sample z-test is a valuable statistical tool for conducting hypothesis testing, and drawing conclusions about population parameters based on the sample data.