This calculator helps you determine whether a sample mean significantly differs from a hypothesized population mean. Perfect for testing if your sample's average differs from an expected value, whether that's comparing test scores to a benchmark, measurements to specifications, or any scenario where you want to know if your data deviates from a known standard.
Ready to test your hypothesis? to explore the 10-step calculation process, or input your own data to discover if your sample truly differs from the expected value.
One-Sample T-Test is a statistical test used to determine whether a sample mean significantly differs from a hypothesized population mean. It's particularly useful when working with small sample sizes and unknown population standard deviation.
Test Statistic:
Where:
Confidence Interval:
Where:
A manufacturer claims their light bulbs last 1000 hours on average. To test this claim:
Calculating t-statistic:
Constructing confidence
Constructing 95% confidence interval:
With degrees of freedom and significance level , we can analyze this result in two ways. First, using the Student's t Distribution Table, we find the critical value for a two-tailed test is ±2.3. Since our calculated t-statistic (-2.00) falls within these critical values, we fail to reject the null hypothesis. Alternatively, using our Student's t Distribution Calculator, we can calculate the two-tailed p-value: . Since this p-value exceeds our significance level of 0.05, we again fail to reject the null hypothesis. Based on this analysis, we conclude there is insufficient evidence to suggest that the true average lifespan of the light bulbs differs from the claimed 1000 hours. Furthermore, we can state with 95% confidence that the true population mean lifespan falls between 970 and 1000 hours.
Cohen's d measures the standardized difference between the sample mean and hypothesized value:
Interpretation guidelines:
To determine required sample size (n) for desired power (1-β):
Where:
Reject H₀ if:
Standard format for scientific reporting:
library(tidyverse)
set.seed(42)
sample_data <- tibble(
value = rnorm(30, mean = 98, sd = 5) # 30 observations, mean=98, sd=5
)
# Hypothesized mean
mu0 <- 100
# Basic summary statistics
summary_stats <- sample_data %>%
summarise(
n = n(),
mean = mean(value),
sd = sd(value),
se = sd/sqrt(n)
)
# One-sample t-test
t_test_result <- t.test(sample_data$value, mu = mu0)
# Effect size (Cohen's d)
cohens_d <- (mean(sample_data$value) - mu0) / sd(sample_data$value)
# Calculate confidence interval
ci <- t_test_result$conf.int
# Print results
print(t_test_result)
print(str_glue("Effect size (Cohen's d):", cohens_d))
# Visualize the data
library(ggplot2)
ggplot(sample_data, aes(x = value)) +
geom_histogram(aes(y = ..density..), bins = 10) +
geom_density() +
geom_vline(xintercept = mu0, color = "red", linetype = "dashed") +
geom_vline(xintercept = mean(sample_data$value), color = "blue") +
theme_minimal() +
labs(title = "Sample Distribution with Hypothesized Mean",
subtitle = "Blue: Sample Mean, Red dashed: Hypothesized Mean")import numpy as np
import scipy.stats as stats
import matplotlib.pyplot as plt
import seaborn as sns
from statsmodels.stats.power import TTestPower
# Generate sample data
np.random.seed(42)
sample_data = np.random.normal(loc=98, scale=5, size=30) # 30 observations, mean=98, sd=5
# Hypothesized mean
mu0 = 100
# Basic summary statistics
n = len(sample_data)
sample_mean = np.mean(sample_data)
sample_sd = np.std(sample_data, ddof=1)
se = sample_sd / np.sqrt(n)
# Perform one-sample t-test
t_stat, p_value = stats.ttest_1samp(sample_data, mu0)
# Calculate Cohen's d effect size
cohens_d = (sample_mean - mu0) / sample_sd
# Calculate confidence interval (95%)
ci = stats.t.interval(confidence=0.95,
df=n-1,
loc=sample_mean,
scale=se)
# Print results
print(f"Sample Statistics:")
print(f"Mean: {sample_mean:.2f}")
print(f"Standard Deviation: {sample_sd:.2f}")
print(f"Standard Error: {se:.2f}")
print(f"nT-Test Results:")
print(f"t-statistic: {t_stat:.2f}")
print(f"p-value: {p_value:.4f}")
print(f"95% CI: [{ci[0]:.2f}, {ci[1]:.2f}]")
print(f"Effect size (Cohen's d): {cohens_d:.2f}")
# Visualize the data
plt.figure(figsize=(10, 6))
sns.histplot(sample_data, stat='density', alpha=0.5)
sns.kdeplot(sample_data)
plt.axvline(mu0, color='red', linestyle='--', label='Hypothesized Mean')
plt.axvline(sample_mean, color='blue', label='Sample Mean')
plt.title('Sample Distribution with Hypothesized Mean')
plt.legend()
plt.show()
# Power analysis
analysis = TTestPower()
power = analysis.power(effect_size=cohens_d,
nobs=n,
alpha=0.05)
print(f"Power Analysis:")
print(f"Statistical Power: {power:.3f}")Consider these alternatives when assumptions are violated:
This calculator helps you determine whether a sample mean significantly differs from a hypothesized population mean. Perfect for testing if your sample's average differs from an expected value, whether that's comparing test scores to a benchmark, measurements to specifications, or any scenario where you want to know if your data deviates from a known standard.
Ready to test your hypothesis? to explore the 10-step calculation process, or input your own data to discover if your sample truly differs from the expected value.
One-Sample T-Test is a statistical test used to determine whether a sample mean significantly differs from a hypothesized population mean. It's particularly useful when working with small sample sizes and unknown population standard deviation.
Test Statistic:
Where:
Confidence Interval:
Where:
A manufacturer claims their light bulbs last 1000 hours on average. To test this claim:
Calculating t-statistic:
Constructing confidence
Constructing 95% confidence interval:
With degrees of freedom and significance level , we can analyze this result in two ways. First, using the Student's t Distribution Table, we find the critical value for a two-tailed test is ±2.3. Since our calculated t-statistic (-2.00) falls within these critical values, we fail to reject the null hypothesis. Alternatively, using our Student's t Distribution Calculator, we can calculate the two-tailed p-value: . Since this p-value exceeds our significance level of 0.05, we again fail to reject the null hypothesis. Based on this analysis, we conclude there is insufficient evidence to suggest that the true average lifespan of the light bulbs differs from the claimed 1000 hours. Furthermore, we can state with 95% confidence that the true population mean lifespan falls between 970 and 1000 hours.
Cohen's d measures the standardized difference between the sample mean and hypothesized value:
Interpretation guidelines:
To determine required sample size (n) for desired power (1-β):
Where:
Reject H₀ if:
Standard format for scientific reporting:
library(tidyverse)
set.seed(42)
sample_data <- tibble(
value = rnorm(30, mean = 98, sd = 5) # 30 observations, mean=98, sd=5
)
# Hypothesized mean
mu0 <- 100
# Basic summary statistics
summary_stats <- sample_data %>%
summarise(
n = n(),
mean = mean(value),
sd = sd(value),
se = sd/sqrt(n)
)
# One-sample t-test
t_test_result <- t.test(sample_data$value, mu = mu0)
# Effect size (Cohen's d)
cohens_d <- (mean(sample_data$value) - mu0) / sd(sample_data$value)
# Calculate confidence interval
ci <- t_test_result$conf.int
# Print results
print(t_test_result)
print(str_glue("Effect size (Cohen's d):", cohens_d))
# Visualize the data
library(ggplot2)
ggplot(sample_data, aes(x = value)) +
geom_histogram(aes(y = ..density..), bins = 10) +
geom_density() +
geom_vline(xintercept = mu0, color = "red", linetype = "dashed") +
geom_vline(xintercept = mean(sample_data$value), color = "blue") +
theme_minimal() +
labs(title = "Sample Distribution with Hypothesized Mean",
subtitle = "Blue: Sample Mean, Red dashed: Hypothesized Mean")import numpy as np
import scipy.stats as stats
import matplotlib.pyplot as plt
import seaborn as sns
from statsmodels.stats.power import TTestPower
# Generate sample data
np.random.seed(42)
sample_data = np.random.normal(loc=98, scale=5, size=30) # 30 observations, mean=98, sd=5
# Hypothesized mean
mu0 = 100
# Basic summary statistics
n = len(sample_data)
sample_mean = np.mean(sample_data)
sample_sd = np.std(sample_data, ddof=1)
se = sample_sd / np.sqrt(n)
# Perform one-sample t-test
t_stat, p_value = stats.ttest_1samp(sample_data, mu0)
# Calculate Cohen's d effect size
cohens_d = (sample_mean - mu0) / sample_sd
# Calculate confidence interval (95%)
ci = stats.t.interval(confidence=0.95,
df=n-1,
loc=sample_mean,
scale=se)
# Print results
print(f"Sample Statistics:")
print(f"Mean: {sample_mean:.2f}")
print(f"Standard Deviation: {sample_sd:.2f}")
print(f"Standard Error: {se:.2f}")
print(f"nT-Test Results:")
print(f"t-statistic: {t_stat:.2f}")
print(f"p-value: {p_value:.4f}")
print(f"95% CI: [{ci[0]:.2f}, {ci[1]:.2f}]")
print(f"Effect size (Cohen's d): {cohens_d:.2f}")
# Visualize the data
plt.figure(figsize=(10, 6))
sns.histplot(sample_data, stat='density', alpha=0.5)
sns.kdeplot(sample_data)
plt.axvline(mu0, color='red', linestyle='--', label='Hypothesized Mean')
plt.axvline(sample_mean, color='blue', label='Sample Mean')
plt.title('Sample Distribution with Hypothesized Mean')
plt.legend()
plt.show()
# Power analysis
analysis = TTestPower()
power = analysis.power(effect_size=cohens_d,
nobs=n,
alpha=0.05)
print(f"Power Analysis:")
print(f"Statistical Power: {power:.3f}")Consider these alternatives when assumptions are violated: