One Sample Z-Test in R

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.

How to 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.

You can find more topics about Z-Score and how to calculate z score given the area on the ZscoreGeek home page.