T-Test Calculator

T-Test is evaluated from Sample Mean, Hypothesized Population Mean and Sample Standard Deviation. The calculation reports t-Statistic, Degrees of Freedom and Standard Error.

Results

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About the T-Test Calculator

T-Test is treated here as a quantitative relation between Sample Mean, Hypothesized Population Mean, Sample Standard Deviation and Sample Size and t-Statistic, Degrees of Freedom, Standard Error and Critical t-Value.

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
t = (sample mean - hypothesized mean) / (sample SD / sqrtn)
Compare |t| to critical value from t-distribution table.

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

t = (sample mean - hypothesized mean) / (sample SD / sqrtn)
Compare |t| to critical value from t-distribution table.

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: Quality control: Is machine producing parts with mean 50mm? Sample: x̄=52.4, s=8.2, n=30

Inputs

sample_mean: 52.4 pop_mean: 50 std_dev: 8.2 n: 30 alpha: 0.05
t-Statistic: 1.6031. Degrees of Freedom: 29. Standard Error: 1.4971. Critical t-Value: 1.9703. Decision: Fail to reject H₀ - not statistically significant

With Sample Mean = 52.4, Hypothesized Population Mean = 50, Sample Standard Deviation = 8.2 and Sample Size = 30 as the stated inputs, the result is t-Statistic = 1.6031, Degrees of Freedom = 29 and Standard Error = 1.4971. Each value corresponds to the declared output fields.

Example 2: Clinical trial: New drug vs. standard 120 mmHg BP. Sample: x̄=115, s=12, n=40

Inputs

sample_mean: 115 pop_mean: 120 std_dev: 12 n: 40 alpha: 0.05
t-Statistic: -2.6352. Degrees of Freedom: 39. Standard Error: 1.8974. Critical t-Value: 1.9677. Decision: Reject H₀ - result is statistically significant

With Sample Mean = 115, Hypothesized Population Mean = 120, Sample Standard Deviation = 12 and Sample Size = 40 as the stated inputs, the result is t-Statistic = -2.6352, Degrees of Freedom = 39 and Standard Error = 1.8974. Each value corresponds to the declared output fields.

Example 3: Education research: New curriculum vs. national average 75. Class scores: x̄=79, s=10, n=25

Inputs

sample_mean: 79 pop_mean: 75 std_dev: 10 n: 25 alpha: 0.05
t-Statistic: 2. Degrees of Freedom: 24. Standard Error: 2. Critical t-Value: 1.9725. Decision: Reject H₀ - result is statistically significant

With Sample Mean = 79, Hypothesized Population Mean = 75, Sample Standard Deviation = 10 and Sample Size = 25 as the stated inputs, the result is t-Statistic = 2, Degrees of Freedom = 24 and Standard Error = 2. Each value corresponds to the declared output fields.

Example 4: Business analytics: Website conversion rate 3.5% vs. industry 3%. n=500 visitors

Inputs

sample_mean: 3.5 pop_mean: 3 std_dev: 1.84 n: 500 alpha: 0.01
t-Statistic: 6.0763. Degrees of Freedom: 499. Standard Error: 0.0823. Critical t-Value: 2.5776. Decision: Reject H₀ - result is statistically significant

With Sample Mean = 3.5, Hypothesized Population Mean = 3, Sample Standard Deviation = 1.84 and Sample Size = 500 as the stated inputs, the result is t-Statistic = 6.0763, Degrees of Freedom = 499 and Standard Error = 0.0823. Each value corresponds to the declared output fields.

Common Use Cases

  • Test if sample mean differs from a known population mean
  • Evaluate whether a process change had a significant effect
  • Compare sample measurement against a standard value