How to Perform One-Proportion Z-Test in R with Practical Examples

Introduction to One-Proportion Z-Tests

The one-proportion Z-test is a statistical method used to test if the observed proportion in a sample differs significantly from a hypothesized population proportion. For example, this test can help determine if the proportion of customers satisfied with a product is statistically different from an industry standard satisfaction rate.

When to Use a One-Proportion Z-Test

The one-proportion Z-test is appropriate when:

• You have a large sample size (n > 30 is recommended).
• You are testing a single sample proportion against a known or hypothesized proportion.
• The data consist of two possible outcomes (e.g., success/failure, yes/no).

Formula for the One-Proportion Z-Test

The Z-test for proportions calculates a z-score that indicates how many standard deviations the observed sample proportion is from the hypothesized proportion. The formula is:

\[ Z = \frac{\hat{p} – p_0}{\sqrt{\frac{p_0 (1 – p_0)}{n}}} \]

Where:

• \( \hat{p} \) = observed sample proportion
• \( p_0 \) = hypothesized population proportion
• \( n \) = sample size

Calculating Left-Tailed, Right-Tailed, and Two-Tailed p-values from the Z-score

After calculating the Z-score, the next step is to determine the p-value , which tells us the probability of observing a result as extreme as the one found, assuming the null hypothesis is true. The type of p-value calculation depends on the direction of the test:

• Left-tailed test: Used when testing if the observed proportion is significantly less than the hypothesized proportion.
• Right-tailed test: Used when testing if the observed proportion is significantly greater than the hypothesized proportion.
• Two-tailed test: Used when testing if the observed proportion is significantly different from the hypothesized proportion, in either direction.

Formulas for Calculating p-values from the Z-score

• Left-tailed p-value: This is the probability that the sample proportion is less than the hypothesized proportion. For a left-tailed test, the p-value is calculated as: \[ p = \text{P(Z ≤ z)} = \text{pnorm}(z) \]
• Right-tailed p-value: This is the probability that the sample proportion is greater than the hypothesized proportion. For a right-tailed test, the p-value is calculated as: \[ p = \text{P(Z ≥ z)} = 1 – \text{pnorm}(z) \]
• Two-tailed p-value: This is the probability that the sample proportion is different from the hypothesized proportion in either direction. For a two-tailed test, the p-value is calculated as: \[ p = 2 \times (1 – \text{pnorm}(|z|)) \]

In R, the pnorm() function is used to calculate the cumulative probability for a given Z-score.

Interpreting the p-values

The p-value helps us determine whether to reject the null hypothesis based on a chosen significance level (typically 0.05):

• If the p-value is less than 0.05 (for the chosen tail direction), we reject the null hypothesis, indicating that the observed proportion is significantly different from the hypothesized proportion in the specified direction.
• If the p-value is greater than 0.05, we fail to reject the null hypothesis, suggesting that the observed proportion is not significantly different from the hypothesized proportion.

By selecting and calculating left-tailed, right-tailed, or two-tailed p-values from the Z-score, we can tailor our analysis to address specific research questions, whether looking for a directional difference or simply any significant deviation.

Calculating the One-Proportion Z-Test in R

In R, we calculate the Z-score manually and use it to find the p-value. Suppose we observe that 45 out of 100 customers are satisfied with a product, and we want to test if this proportion (0.45) is significantly different from the hypothesized industry standard of 0.50.

Example Calculation

Using the example data where 45 out of 100 customers are satisfied (observed proportion = 0.45) and we are testing against a hypothesized proportion of 0.50, the Z-score calculation and p-value interpretation are as follows:

# Given values
observed_successes <- 45
sample_size <- 100
sample_proportion <- observed_successes / sample_size  # Observed proportion
hypothesized_proportion <- 0.50  # Hypothesized proportion

# Calculate the Z-score
z_score <- (sample_proportion - hypothesized_proportion) / sqrt(hypothesized_proportion * (1 - hypothesized_proportion) / sample_size)
z_score

# Calculate the p-value for a two-tailed test
p_value <- 2 * (1 - pnorm(abs(z_score)))
p_value

The output will display the Z-score and p-value:

# Expected Output:
Z-score: -1
p-value: 0.3173105

Explanation

This output provides the following insights:

• Z-score: The Z-score of -1 indicates that the observed proportion (0.45) is 1 standard deviation below the hypothesized proportion (0.50).
• p-value: The p-value of 0.3173 is greater than the common significance level of 0.05, so we fail to reject the null hypothesis. This indicates that there is not enough evidence to conclude that the observed proportion (0.45) differs significantly from the hypothesized proportion (0.50).

Interpreting the Results

Suppose the calculated Z-score is -1, and the p-value is 0.3173. Here’s how to interpret these results:

• The Z-score of -1 suggests that the observed proportion is 1 standard deviation below the hypothesized proportion of 0.50.
• The p-value of 0.3173 is greater than the common significance level of 0.05, so we fail to reject the null hypothesis. This indicates that there is not enough evidence to conclude that the observed proportion (0.45) differs significantly from the hypothesized proportion (0.50).

Assumptions and Limitations

The one-proportion Z-test relies on certain assumptions and has limitations to be aware of:

Assumptions

• Large Sample Size: For the one-proportion Z-test to be accurate, we generally need a large sample size (typically n > 30). This large sample size allows us to use the normal distribution as an approximation of the binomial distribution, which is the actual distribution of proportions. In statistical terms, this is often called the "normal approximation." When the sample size is large, the Central Limit Theorem tells us that the distribution of the sample proportion will approximate a normal distribution, making it appropriate to use the Z-test. However, with smaller samples, this approximation may not hold, and results may be less reliable.
• Independent Observations: The Z-test assumes that each observation in the sample is independent, meaning that one outcome does not affect another. For example, if you are measuring customer satisfaction, each customer's response should be independent of others. This assumption is crucial because dependencies between observations can lead to biased results and inaccurate conclusions.
• Binary Outcomes: The one-proportion Z-test applies when we have data with two possible outcomes, often called "success" and "failure." For instance, if you are looking at customer satisfaction, each response could be either "satisfied" (success) or "not satisfied" (failure). Having just two possible outcomes allows us to use the binomial distribution, which is necessary for the Z-test to be valid for proportions.

Limitations

• Not Suitable for Small Samples: The Z-test is less reliable for small samples. When the sample size is small, the binomial distribution (rather than the normal distribution) should ideally be used to represent the proportion. In such cases, a binomial test, which does not rely on the normal approximation, is often preferred because it directly calculates probabilities based on the binomial distribution. The binomial test is generally more accurate for small sample sizes or when dealing with rare events.
• Sensitivity to Proportion Values: The accuracy of the Z-test can depend on the hypothesized proportion (\( p_0 \)). When this proportion is close to 0 or 1, the normal approximation may break down, making the Z-test less reliable. For example, if we hypothesize that 98% of people prefer a specific brand, even small sample deviations can create inaccurate conclusions. In these cases, we must interpret Z-test results with caution or consider alternative methods.
• Reliance on Approximation: The one-proportion Z-test relies on the normal approximation of the binomial distribution. This approximation can become unreliable if the sample size is too small or if the observed and hypothesized proportions are very different. In other words, if the data do not approximate a bell-shaped (normal) curve, the Z-test’s results might not accurately reflect the true nature of the data. It’s essential to check that the sample size and proportion are appropriate before applying the Z-test to ensure accurate results.

Conclusion

The one-proportion Z-test is a useful tool for determining whether an observed sample proportion differs significantly from a hypothesized population proportion. By calculating the Z-score and interpreting the p-value, we can assess whether the observed proportion is likely due to random chance or represents a significant deviation from the hypothesized value.

Although the Z-test is straightforward, it has certain limitations, particularly with small samples or extreme proportions. Understanding these limitations ensures that the Z-test is applied accurately and that results are interpreted with caution. For cases where assumptions are not met, alternative methods such as the binomial test may be more suitable.

Try the One-Proportion Z-Test Calculator

For an easy way to calculate the one-proportion Z-test and determine the p-value, you can check out our One-Proportion Z-Test Calculator on the Research Scientist Pod.

Have fun and happy researching!