Quick Answer: Type I and Type II Error Result

The verified SPSS output uses G3 final grade as the observed test variable and compares it against the null test value of 10. The descriptive output shows N = 649, mean = 11.91, standard deviation = 3.231, and standard error = .127. The one-sample t-test output shows t(648) = 15.030, SPSS Sig. = .000, mean difference = 1.906, and 95% confidence interval for the difference from 1.66 to 2.16. The Python p-value decision chart gives the exact observed p-value as 5.068e-44, which is far below alpha = .05.

Hypothesis-style decision: The null hypothesis says the population mean equals the test value, meaning H₀: μ = 10. The alternative hypothesis says the population mean differs from the test value, meaning H₁: μ ≠ 10. Since p = 5.068e-44 < .05, the correct decision is to reject H₀. In this decision framework, a Type I Error would mean rejecting H₀ if H₀ were actually true, while a Type II Error would mean failing to reject H₀ if a real difference existed.

What Are Type I and Type II Errors?

Type I and Type II Error are decision errors that can happen in hypothesis testing. A hypothesis test compares sample evidence with a null hypothesis. The researcher then decides whether to reject H₀ or fail to reject H₀. Because the decision is based on sample data, it is possible to make the wrong decision.

A Type I Error happens when the null hypothesis is true but the test rejects it. This is called a false positive. In practical terms, it means the test claims that an effect, difference, relationship, or change exists when it does not really exist in the population. The long-run probability of Type I Error is controlled by the significance level, called alpha (α).

A Type II Error happens when the null hypothesis is false but the test fails to reject it. This is called a false negative. In practical terms, it means the test misses a real effect, difference, relationship, or change. The probability of Type II Error is called beta (β). The probability of correctly detecting a real effect is called statistical power, and it equals 1 − β.

Type I Error, Type II Error, Alpha, Beta and Power Formulas

The Type I and Type II Error framework is built from decision probabilities. The formulas are simple but very important for interpreting statistical tests.

Alpha is selected before testing. A common value is α = .05. This means the test rule allows a 5% long-run false-positive rate if the null hypothesis is actually true. Beta is usually evaluated through power analysis. If power is .80, then beta is .20. If power is .95, then beta is .05.

Null and Alternative Hypothesis for This Example

The SPSS output uses a one-sample test context where G3 final grade is compared against the test value of 10. The observed sample mean is 11.91. The charted null distribution is centered near 10, and the observed mean is far to the right of the null value.

Dataset and Test Variables Used

The worked example uses the student performance data structure and focuses on G3 final grade. The one-sample test compares the observed G3 mean against the null test value of 10. The SPSS output also includes a group example by sex and a correlation example between G2 and G3 to show how decision errors apply across different hypothesis-testing contexts.

For broader reporting context, use descriptive statistics, confidence interval, effect size, and cross tabulation.

Verified SPSS Output Interpretation

The SPSS output verifies the observed test context, one-sample hypothesis decision, decision matrix, group test example, and correlation example. These outputs help explain how Type I and Type II Error logic applies across common statistical procedures.

SPSS Descriptive Output for G3

SPSS One-Sample t-test Output

SPSS Group Test Example

The SPSS output also provides an independent-samples example comparing G3 by sex. Female students had n = 383, M = 12.25, SD = 3.124. Male students had n = 266, M = 11.41, SD = 3.321. Levene’s test was not significant, F = .004, p = .950, supporting the equal-variance row. The equal-variance t-test showed t(647) = 3.311, p = .001, mean difference = .847, and 95% CI from .345 to 1.350.

SPSS Correlation Example

The SPSS output also shows a strong correlation between G2 and G3, with r = .919, p < .001, and N = 649. In a correlation test, a Type I Error would mean concluding that a population correlation exists when it does not. A Type II Error would mean failing to detect a real population correlation. Because the observed relationship is very strong and significant, the result demonstrates a low-risk detection example under this sample size.

Python Chart-by-Chart Interpretation

The Python charts explain Type I and Type II Error visually. The main message is consistent across the figures: a Type I Error is a false positive, a Type II Error is a false negative, alpha controls false-positive risk, beta represents false-negative risk, and power improves when sample size and effect size increase.

Python Chart 1: Type I and Type II Error Decision Matrix

This decision matrix is the clearest visual summary of Type I and Type II Error. The x-axis shows the test decision: fail to reject H₀ or reject H₀. The y-axis shows reality: H₀ true or H₁ true. If H₀ is true and the test fails to reject it, the decision is correct. If H₀ is true and the test rejects it, that is a Type I Error, or false positive.

The lower row explains the alternative case. If H₁ is true and the test fails to reject H₀, that is a Type II Error, or false negative. If H₁ is true and the test rejects H₀, the decision is correct and represents statistical power. This chart is useful because it shows that a wrong decision is not determined by the test decision alone. The same decision can be correct or incorrect depending on the true state of reality.

Python Chart 2: Null and Alternative Distributions with Alpha and Beta Regions

This chart explains Type I and Type II Error using probability distributions. The blue curve represents the null hypothesis distribution, centered near the null value of 10. The orange curve represents the alternative distribution based on the observed mean near 11.91. The shaded null tails represent alpha, which is the probability of rejecting H₀ when H₀ is actually true. That is the Type I Error region.

The alternative distribution shows how Type II Error occurs. If the alternative is true but the observed test result falls inside the non-rejection region, the test fails to detect the real difference. That area is beta. In this example, the observed mean is far from the null center, so the test has very strong evidence against H₀. The separation between the null and alternative curves explains why the p-value is extremely small.

Python Chart 3: Alpha, Beta and Power Tradeoff

This chart explains the relationship among alpha, beta and power. The x-axis displays alpha levels, which represent the Type I Error rate. The y-axis displays probability. In this worked example, power remains near 1.00 and beta remains near 0.00 across the alpha levels because the observed effect is very strong and the sample size is large. This means the test is highly likely to detect the effect in this example.

The important conceptual lesson is that lowering alpha can reduce false positives, but it may increase false negatives when the sample size or effect size is not large enough. In weak-effect or small-sample research, stricter alpha makes it harder to reject H₀, which can increase Type II Error risk. This chart should therefore be read as a demonstration of a strong-effect case, not as a claim that beta is always zero.

Python Chart 4: Type II Error Risk, Sample Size and Effect Size

This heatmap is the most practical chart for Type II Error. It shows how power changes across sample sizes and effect sizes. When the effect size is very small, such as d = 0.1, power is low at small sample sizes. At n = 20, power is only 0.07. Even at n = 500, power reaches only 0.61. This means small effects are difficult to detect, so Type II Error risk remains high unless the sample size is very large.

For d = 0.2, power improves from 0.15 at n = 20 to 0.99 at n = 500. For d = 0.3, power is already 0.85 at n = 100. For d = 0.5 and d = 0.8, power becomes very high quickly. This chart proves why power analysis is essential: low-powered studies can miss real effects, creating Type II Errors.

Python Chart 5: Type I Error Rule, p-value and Alpha

This chart gives the final hypothesis-test decision. The alpha threshold is 0.05, and the observed p-value is 5.068e-44. Because the p-value is far below alpha, the decision is to reject H₀. The chart also explains Type I Error because rejecting H₀ at alpha = .05 accepts a 5% long-run false-positive risk if H₀ is actually true.

For this specific example, the p-value is so small that the evidence against the null hypothesis is extremely strong. However, the correct wording is still based on probability and decision rules. We do not say the null is impossible. We say the sample evidence is strong enough to reject H₀ at the selected alpha level.

R Chart-by-Chart Validation

The R charts validate the same Type I and Type II Error logic using a separate software workflow. The R output confirms the decision matrix, alpha-beta distribution logic, power demonstration, sample-size/effect-size heatmap and p-value decision rule.

R Chart 1: Type I and Type II Error Decision Matrix

The R decision matrix confirms the same four-cell structure. If H₀ is true and the test fails to reject it, the decision is correct. If H₀ is true and the test rejects it, the decision is a Type I Error. If H₁ is true and the test fails to reject H₀, the decision is a Type II Error. If H₁ is true and the test rejects H₀, the decision is correct and represents power.

R Chart 2: Null and Alternative Distributions

The R distribution chart confirms that alpha is located in the rejection tails under the null distribution, while beta is located in the non-rejection region under the alternative distribution. The greater the separation between the null and alternative distributions, the smaller beta becomes and the higher power becomes.

R Chart 3: Alpha, Beta and Power Tradeoff

The R alpha-beta-power chart validates the Python interpretation. In this strong-effect example, power remains near 1 and beta remains near 0. This happens because the observed mean is far from the null value. In weaker studies, the tradeoff between alpha and beta becomes more visible.

R Chart 4: Power Grid and Type II Error Risk

The R heatmap confirms the practical power lesson. Small effects require large samples to avoid Type II Error. Large effects require fewer cases to reach high power. This chart supports the recommendation that power analysis should be done before data collection whenever possible.

R Chart 5: p-value and Alpha Decision Rule

The R p-value decision chart confirms the final conclusion. The observed p-value is far below alpha, so H₀ is rejected. This chart should be used in reports to explain the decision rule clearly: compare the p-value with alpha, then decide whether to reject the null hypothesis.

SPSS, R, Python and Excel Workflows for Type I and Type II Error

The same Type I and Type II Error logic can be applied in SPSS, R, Python and Excel. SPSS provides the test output and p-value. Python and R can calculate the test result, draw charts and estimate power. Excel can calculate the p-value and decision rule manually.

SPSS Workflow

R Workflow

Python Workflow

Excel Workflow

Code Blocks for Type I and Type II Error

SPSS Syntax for One-Sample Test and Type I / Type II Error Context

* Type I and Type II Error demonstration in SPSS.
* Main observed variable: G3 final grade.
* Null test value: 10.

TITLE "Type I and Type II Error: One-Sample Test Context".

DESCRIPTIVES VARIABLES=G3
  /STATISTICS=MEAN STDDEV MIN MAX.

T-TEST
  /TESTVAL=10
  /MISSING=ANALYSIS
  /VARIABLES=G3
  /CRITERIA=CI(.95).

* Optional group example: G3 by sex.
T-TEST GROUPS=sex('F' 'M')
  /MISSING=ANALYSIS
  /VARIABLES=G3
  /CRITERIA=CI(.95).

* Optional correlation example: G2 and G3.
CORRELATIONS
  /VARIABLES=G2 G3
  /PRINT=TWOTAIL NOSIG
  /MISSING=PAIRWISE.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE="Type-I-and-Type-II-Error-SPSS-Output.pdf".

Python Code for Type I and Type II Error

import pandas as pd
import numpy as np
from scipy import stats
from statsmodels.stats.power import TTestPower

df = pd.read_csv("dataset.csv")

x = pd.to_numeric(df["G3"], errors="coerce").dropna()

null_value = 10
alpha = 0.05

t_stat, p_value = stats.ttest_1samp(x, popmean=null_value)

mean_value = x.mean()
sd_value = x.std(ddof=1)
se_value = sd_value / np.sqrt(len(x))
mean_diff = mean_value - null_value

print("N:", len(x))
print("Mean:", mean_value)
print("SD:", sd_value)
print("SE:", se_value)
print("Mean difference:", mean_diff)
print("t statistic:", t_stat)
print("p-value:", p_value)

if p_value <= alpha:
    print("Reject H0")
else:
    print("Fail to reject H0")

# Power examples for effect sizes and sample sizes
analysis = TTestPower()

sample_sizes = [20, 40, 60, 100, 150, 200, 300, 500]
effect_sizes = [0.1, 0.2, 0.3, 0.5, 0.8]

for d in effect_sizes:
    for n in sample_sizes:
        power = analysis.power(effect_size=d, nobs=n, alpha=alpha, alternative="two-sided")
        beta = 1 - power
        print("effect size:", d, "n:", n, "power:", round(power, 3), "beta:", round(beta, 3))

R Code for Type I and Type II Error

# Type I and Type II Error demonstration in R

df <- read.csv("dataset.csv")

x <- as.numeric(df$G3)
x <- x[!is.na(x)]

null_value <- 10
alpha <- 0.05

result <- t.test(x, mu = null_value, alternative = "two.sided")
print(result)

if(result$p.value <= alpha){
  print("Reject H0")
} else {
  print("Fail to reject H0")
}

# Power analysis
# install.packages("pwr")
library(pwr)

sample_sizes <- c(20, 40, 60, 100, 150, 200, 300, 500)
effect_sizes <- c(0.1, 0.2, 0.3, 0.5, 0.8)

power_table <- expand.grid(effect_size = effect_sizes, n = sample_sizes)
power_table$power <- mapply(function(d, n){
  pwr.t.test(n = n, d = d, sig.level = alpha, type = "one.sample", alternative = "two.sided")$power
}, power_table$effect_size, power_table$n)

power_table$beta <- 1 - power_table$power
print(power_table)

Excel Formulas for Type I and Type II Error

Assume G3 values are in A2:A650.

Sample size:
=COUNT(A2:A650)

Mean:
=AVERAGE(A2:A650)

Standard deviation:
=STDEV.S(A2:A650)

Standard error:
=STDEV.S(A2:A650)/SQRT(COUNT(A2:A650))

Null test value:
10

t statistic:
=(AVERAGE(A2:A650)-10)/(STDEV.S(A2:A650)/SQRT(COUNT(A2:A650)))

Degrees of freedom:
=COUNT(A2:A650)-1

Two-tailed p-value:
=T.DIST.2T(ABS(t_cell), df_cell)

Decision:
=IF(p_cell<=0.05,"Reject H0","Fail to reject H0")

Type I Error meaning:
Rejecting H0 when H0 is actually true.

Type II Error meaning:
Failing to reject H0 when H1 is actually true.

Power:
Power = 1 - Beta

APA Reporting Wording for Type I and Type II Error

When reporting Type I and Type II Error, do not only define the terms. Connect them to the actual hypothesis test, p-value, alpha level and power interpretation.

APA-Style One-Sample Test Report

APA-Style Group Test Report

Student-Friendly Report Example

Common Mistakes in Type I and Type II Error Interpretation

When to Use Type I and Type II Error Concepts

Use Type I and Type II Error concepts whenever you interpret a hypothesis test. These concepts are especially important when decisions have real consequences, such as clinical trials, educational interventions, quality control, policy evaluation, medical screening and business testing.

Type I and Type II Error should be discussed together with confidence intervals, effect size, and assumption checks such as Levene test, Brown-Forsythe test, and Q-Q plot normality check.

Downloads and Resources for Type I and Type II Error

The resources below include the SPSS output PDF, Python charts, and R validation charts used in this guide.

Related Statistical Guides

Use these related guides to connect Type I and Type II Error with hypothesis testing, assumption testing, normality checks, variance checks, confidence intervals and reporting.