One Proportion Z Test: Formula, Example, Calculator, SPSS, R, Python and Excel improvement Guide 1

What Is a One Proportion Z Test?

A One Proportion Z Test is a hypothesis test for a single population proportion. It is used when the outcome has two categories, such as yes/no, pass/fail, success/failure, present/absent or support/not support. The test asks whether the observed sample proportion is far enough away from a hypothesized value to reject the null hypothesis.

In this example, the binary response is higher. Students answered whether they wanted higher education. The success category is yes. The statistical question is: Is the proportion of students who want higher education greater than 80%?

The test is called a z test because it converts the difference between the observed proportion and the hypothesized proportion into a standard normal z statistic. When the sample is large enough, the sampling distribution of the sample proportion can be approximated by the normal distribution. That is why the test uses the standard normal curve to calculate the p-value.

When Should You Use a One Proportion Z Test?

Use a One Proportion Z Test when you have one sample, one binary outcome and one hypothesized population proportion. It is common in education, public health, market research, survey analysis, election polling, quality control and website conversion analysis.

One Proportion Z Test Hypotheses and Formula

The One Proportion Z Test formula compares the observed sample proportion to the hypothesized null proportion. The formula is:

z = (p-hat - p0) / sqrt[p0(1 - p0) / n]

Here, p-hat is the observed sample proportion, p0 is the hypothesized population proportion under the null hypothesis, and n is the sample size. The denominator is the standard error under the null hypothesis.

Hypotheses for This Worked Example

Conditions for One Proportion Z Test

The conditions for One Proportion Z Test must be checked before relying on the normal approximation. The outcome should be binary, the observations should be independent, and the expected success and failure counts under the null hypothesis should usually be at least 10.

Dataset and Variable Used

This tutorial uses the student-por.csv student performance dataset. The main variable for the one proportion z test is higher. The response yes is coded as the success category, meaning the student wants higher education.

External dataset source: UCI Machine Learning Repository: Student Performance dataset.

Verified R, Python and SPSS Results

The One Proportion Z Test was reproduced in R, Python and SPSS. The fixed SPSS output confirms the same result as the R and Python workflows.

Main One Proportion Z Test Result

Confidence Intervals

Descriptive Group Context

R Charts for One Proportion Z Test

The R workflow produced visual explanations of the observed proportion, hypothesized proportion, success/failure counts, confidence interval, z statistic, large-sample condition and p-value.

1. Observed vs Hypothesized Proportion

This chart gives the simplest interpretation of the test. The observed sample proportion is clearly higher than the hypothesized 80% benchmark. The statistical test then asks whether this difference is large enough after considering sample size and standard error.

2. Success and Failure Counts

The count chart shows that most students answered yes. Since 580 out of 649 students selected yes, the observed sample proportion is 580 / 649 = .8937.

3. Confidence Interval vs Null Proportion

The confidence interval is above the 80% benchmark. This supports the same conclusion as the z test: the population proportion is likely greater than .80.

4. Z Statistic Normal Curve

The z statistic is 5.967, which is far to the right of the standard normal center. This creates a very small right-tailed p-value. Therefore, the result is reported as p < .001.

5. Large-Sample Condition Check

The large-sample condition is strongly satisfied. Both expected counts are much greater than the usual minimum rule of 10, so the normal approximation is appropriate.

6. Higher Education Aspiration by School

The school chart provides descriptive context. GP has 391 yes responses out of 423 students, while MS has 189 yes responses out of 226 students. This is not the main z test, but it helps readers understand how the yes responses are distributed across groups.

7. Higher Education Aspiration by Sex

The sex chart shows that both female and male students have high yes proportions. Female students show 348/383 = 90.9%, while male students show 232/266 = 87.2%.

8. P-Value Summary

The main one-sided p-value is below .001, which is far smaller than alpha = .05. This supports rejecting the null hypothesis.

9. Formula Summary Panel

The formula panel is useful for students because it shows the calculation in one place. It connects the raw count, sample size, observed proportion and null proportion to the final z statistic.

Python Validation Charts for One Proportion Z Test

The Python workflow independently validates the same result. It reproduces the observed proportion, confidence interval, z statistic, p-value summary, group context and formula calculation.

1. Python Observed vs Hypothesized Proportion

The Python chart confirms that the observed sample proportion is clearly above the hypothesized proportion. This is the visual reason the right-tailed test becomes significant.

2. Python Success and Failure Counts

The Python count chart confirms the same raw frequency table used in SPSS and R: 580 yes responses and 69 no responses.

3. Python Confidence Interval

The Wilson 95% confidence interval is approximately [.8676, .9151]. Since the whole interval is above .80, it supports the conclusion that the proportion is greater than the hypothesized value.

4. Python Z Statistic Normal Curve

The z statistic is far into the right tail of the standard normal curve. This explains why the right-tailed p-value is below .001.

5. Python Candidate Binary Variables

This chart shows why higher is a strong example for a one proportion z test. It has the highest yes proportion among the binary variables examined, with 89.4% yes responses.

6. Python Large-Sample Condition Check

The large-sample chart confirms that the normal approximation is safe. The expected success count is 519.2 and the expected failure count is 129.8.

7. Python Higher Education Aspiration by School

The Python school chart confirms the same descriptive group values: GP = 92.4% and MS = 83.6%.

8. Python Higher Education Aspiration by Sex

The Python sex chart confirms female students at 90.9% and male students at 87.2%. Both are high, but these group values are descriptive rather than the main hypothesis test.

9. Python P-Value Summary

The p-value chart shows that the p-value is much smaller than the alpha level. This makes the decision to reject H0 straightforward.

10. Python Formula Summary

The Python formula panel confirms every numerical component of the test. The key values are x = 580, n = 649, p-hat = .8937, p0 = .8000, SE = .01570 and z = 5.967.

How to Run One Proportion Z Test in SPSS, R, Python and Excel

One Proportion Z Test in SPSS

SPSS does not always provide a direct menu-based one proportion z test table in the same way that R or Python can calculate it manually. The clean approach is to create a binary success variable, aggregate the success count and sample size, and calculate p-hat, standard error, z statistic and p-value using SPSS syntax.

* One Proportion Z Test in SPSS.

GET DATA
 /TYPE=TXT
 /FILE='D:\low kda score priority basis posts\first post\One Proportion Z Test\student-por.csv'
 /ENCODING='UTF8'
 /DELCASE=LINE
 /DELIMITERS=","
 /QUALIFIER='"'
 /ARRANGEMENT=DELIMITED
 /FIRSTCASE=2
 /VARIABLES=
 school A40 sex A20 age F8.0 address A20 famsize A20 Pstatus A20
 Medu F8.0 Fedu F8.0 Mjob A40 Fjob A40 reason A40 guardian A40
 traveltime F8.0 studytime F8.0 failures F8.0 schoolsup A40 famsup A40
 paid A40 activities A40 nursery A40 higher A40 internet A40 romantic A40
 famrel F8.0 freetime F8.0 goout F8.0 Dalc F8.0 Walc F8.0
 health F8.0 absences F8.0 G1 F8.0 G2 F8.0 G3 F8.0.
EXECUTE.

STRING higher_clean (A40).
COMPUTE higher_clean = RTRIM(LTRIM(higher)).
COMPUTE success = 0.
IF (higher_clean = "yes") success = 1.
IF (higher_clean = "no") success = 0.
EXECUTE.

AGGREGATE
 /OUTFILE=* MODE=ADDVARIABLES
 /BREAK=
 /n_valid = N(success)
 /x_success = SUM(success).

COMPUTE p0 = .80.
COMPUTE p_hat = x_success / n_valid.
COMPUTE se_h0 = SQRT((p0 * (1 - p0)) / n_valid).
COMPUTE z_stat = (p_hat - p0) / se_h0.
COMPUTE p_value_right_tailed = 1 - CDF.NORMAL(z_stat, 0, 1).
COMPUTE p_value_two_sided = 2 * (1 - CDF.NORMAL(ABS(z_stat), 0, 1)).
EXECUTE.

TEMPORARY.
SELECT IF ($CASENUM = 1).
LIST VARIABLES=n_valid x_success p_hat p0 se_h0 z_stat p_value_right_tailed p_value_two_sided
 /CASES=FROM 1 TO 1.

One Proportion Z Test in R

In R, the calculation can be done manually using the formula. This gives full control over one-sided and two-sided p-values.

student <- read.csv("student-por.csv")

success <- ifelse(trimws(tolower(student$higher)) == "yes", 1, 0)

n <- length(success)
x <- sum(success)
p_hat <- x / n
p0 <- 0.80

se_h0 <- sqrt(p0 * (1 - p0) / n)
z_value <- (p_hat - p0) / se_h0

p_value_right <- pnorm(z_value, lower.tail = FALSE)
p_value_two_sided <- 2 * pnorm(-abs(z_value))

n
x
p_hat
z_value
p_value_right
p_value_two_sided

One Proportion Z Test in Python

Python can reproduce the same test using pandas for data handling and SciPy for the standard normal p-value.

import pandas as pd
import math
from scipy.stats import norm

student = pd.read_csv("student-por.csv")

success = (student["higher"].str.strip().str.lower() == "yes").astype(int)

n = len(success)
x = success.sum()
p_hat = x / n
p0 = 0.80

se_h0 = math.sqrt(p0 * (1 - p0) / n)
z_value = (p_hat - p0) / se_h0

p_value_right = norm.sf(z_value)
p_value_two_sided = 2 * norm.sf(abs(z_value))

print(n, x, p_hat, se_h0, z_value, p_value_right, p_value_two_sided)

One Proportion Z Test in Excel

Excel is useful for students who want to understand the calculator logic behind the test. Suppose column A contains the higher responses from row 2 to row 650.

One Proportion Z Test Calculator Logic

A One Proportion Z Test calculator usually asks for four inputs: success count, sample size, hypothesized proportion and alternative direction. For this example, the inputs are:

The calculator should return approximately p-hat = .8937, SE = .01570, z = 5.967 and p < .001.

How to Report One Proportion Z Test

A good report includes the test name, sample size, success count, observed proportion, hypothesized proportion, z statistic, p-value and decision.

Common Mistakes in One Proportion Z Test

1. Using the test for numeric means

The One Proportion Z Test is for binary outcomes, not continuous variables. For numeric means, use a one-sample t test or z test for a mean.

2. Forgetting the large-sample condition

The normal approximation should be used only when the expected success and failure counts are large enough. In this example, n × p0 and n × (1 − p0) are both much larger than 10.

3. Mixing up p-hat and p0

p-hat is the observed sample proportion. p0 is the hypothesized proportion under the null hypothesis.

4. Reporting p = .000

Statistical software may print .000 for very small p-values. Report it as p < .001.

5. Using a two-sided p-value when the hypothesis is one-sided

If the research question is specifically whether the proportion is greater than p0, the right-tailed p-value is the main p-value. In this article, the main alternative is p > .80.

Related Data Analysis Guides

For more statistical test tutorials, read related guides on Online Internet Cafe. If you are checking normality, see the Kolmogorov-Smirnov Test, Lilliefors Test, DAgostino Pearson Test and Cramer von Mises Test. If you are checking variance assumptions, see the Brown-Forsythe Test, Hartley F Max Test and Cochran C Test.

Download SPSS Output and Verification Files

The fixed SPSS output verifies the correct success count, observed proportion, one-proportion z statistic, p-value, confidence intervals and descriptive crosstab context.

External References

This guide uses verified R, Python and SPSS outputs together with standard statistical documentation and software references. These resources help readers understand proportion testing, z statistics, p-values and confidence intervals.

• UCI Machine Learning Repository: Student Performance dataset
• IBM SPSS Statistics documentation
• The Comprehensive R Archive Network
• SciPy normal distribution documentation
• Microsoft Excel support documentation