T-Score to P-Value Calculator

Understanding the T-Score and P-Value

1. What is a T-Score?

The T-Score measures how far a sample mean is from the population mean in units of standard error. It is calculated as:

\[ T = \frac{\bar{X} - \mu}{\frac{s}{\sqrt{n}}} \]

where:

• \( \bar{X} \): sample mean
• \( \mu \): population mean under the null hypothesis
• \( s \): sample standard deviation
• \( n \): sample size

2. Converting T-Scores to P-Values

The P-value represents the probability of observing a test result as extreme as, or more extreme than, the observed value, assuming the null hypothesis is true. The P-value calculation depends on whether the test is one-tailed or two-tailed:

• One-tailed test: The P-value is the area to the right (or left) of the observed t-score.
• Two-tailed test: The P-value is twice the area to the right of the absolute t-score, accounting for both tails.

The cumulative density function (CDF) of the t-distribution, denoted as \( t_{\text{cdf}} \), is used to find this probability:

\[ \text{One-tailed P-value} = 1 - t_{\text{cdf}}(|T|, \text{df}) \] \[ \text{Two-tailed P-value} = 2 \times (1 - t_{\text{cdf}}(|T|, \text{df})) \]

Finding the CDF of the T-Distribution

The CDF of the t-distribution can be found in several ways:

• Using Statistical Tables: T-distribution tables often provide cumulative probabilities for various t-scores and degrees of freedom. Locate your degrees of freedom in the table and find the cumulative probability associated with your t-score.
• Using Software or Online Tools: You can use statistical software like R, Python, or online calculators. In Python, for instance, the SciPy library offers a straightforward method: from scipy.stats import t p_value = t.cdf(t_score, df=degrees_of_freedom) This line returns the CDF (probability) for a given t-score and degrees of freedom.
• Using JavaScript: Libraries like jStat offer functions to calculate the CDF directly. In JavaScript, you would write: let p_value = jStat.studentt.cdf(tScore, degreesOfFreedom); This will give you the cumulative probability for a t-score with specified degrees of freedom.

3. Interpreting the Results

In hypothesis testing, we compare the P-value to a chosen significance level (α):

• If \( \text{P-value} < \alpha \): We reject the null hypothesis (H₀).
• If \( \text{P-value} \geq \alpha \): We fail to reject the null hypothesis (H₀).

Common significance levels are 0.05 and 0.01, indicating a 5% or 1% threshold for rejecting the null hypothesis.

Real-Life Example: Testing a New Medication

Suppose a pharmaceutical company is testing a new medication to reduce blood pressure. They conduct an experiment with:

• Medication group (mean reduction of 8 mm Hg, sample standard deviation of 3 mm Hg, sample size of 30)
• Placebo group with a hypothesized mean of 5 mm Hg reduction.

The company uses a t-test to check if the medication group has a significantly lower blood pressure than the placebo.

One-Tailed Example:

Testing if the medication group shows a significant reduction in blood pressure:

Two-Tailed Example:

Testing if the medication has a different effect (increase or decrease) compared to the placebo: