One Sample T-Statistic Calculator

One Sample T-Statistic Formula:

\[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \]

T-Statistic (t):

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1. What is the One Sample T-Statistic?

The one sample t-statistic measures how many standard errors the sample mean is from the hypothesized population mean. It's used in hypothesis testing to determine if a sample comes from a population with a specific mean.

2. How Does the Calculator Work?

The calculator uses the one sample t-statistic formula:

\[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \]

Where:

\( \bar{x} \) — Sample mean
\( \mu_0 \) — Hypothesized population mean
\( s \) — Sample standard deviation
\( n \) — Sample size

Explanation: The numerator measures how far the sample mean is from the hypothesized mean, while the denominator (standard error) scales this difference by the sample size.

3. Importance of T-Statistic

Details: The t-statistic is crucial for conducting t-tests, which help determine if observed differences between sample and population means are statistically significant or due to random chance.

4. Using the Calculator

Tips: Enter the sample mean, hypothesized mean, sample standard deviation (must be ≥0), and sample size (must be ≥1). All values must be valid numbers.

5. Frequently Asked Questions (FAQ)

Q1: When should I use a one sample t-test?
A: When you want to compare a sample mean to a known or hypothesized population mean, especially when the population standard deviation is unknown and sample size is small (typically n < 30).

Q2: What does the t-statistic value mean?
A: Larger absolute values indicate greater deviation from the hypothesized mean. The sign indicates direction (positive = sample mean > hypothesized mean).

Q3: How is this different from a z-test?
A: t-tests are used when population standard deviation is unknown (using sample SD instead) and with small samples. They have heavier tails than the normal distribution used in z-tests.

Q4: What are the assumptions of this test?
A: The data should be approximately normally distributed, observations should be independent, and the sample should be randomly selected.

Q5: How do I interpret the t-statistic?
A: Compare it to critical values from the t-distribution (with n-1 degrees of freedom) to determine statistical significance at your chosen alpha level.