T.DIST

Calculates the Student’s left-tailed t-distribution, commonly known as left-tailed t-distribution.

Syntax:

T.DIST(x, deg_freedom, cumulative)

x is required, and is the value that you want to use to evaluate the distribution.

deg_freedom is required, and is the degrees of freedom of the distribution.

cumulative is required, and is a logical value:

TRUE: Returns the cumulative distribution function (CDF) of the left-tailed t-distribution up to the given x value.

FALSE: Returns the probability density function (PDF) at the given x value.

Example:

If x contains 2.1, deg_freedom contains 19 and cumulative contains TRUE:

T.DIST(2.1, 19, TRUE)

returns 0.975342801

x:

deg_freedom:

Cumulative:

Result:

0.975342801

Application:

Scenario

A quality control engineer at a factory wants to determine if a new manufacturing process for a certain part is producing parts with a mean weight of 10 grams. The engineer takes a random sample of 15 parts from the new process and measures their weights. The sample mean is found to be 9.8 grams, and the sample standard deviation is 0.5 grams.

The engineer wants to test the hypothesis that the new process produces parts with a mean weight different from 10 grams. To do this, they will calculate the probability of observing a sample mean of 9.8 grams or less, assuming the true mean is 10 grams.

Step 1: Formulate the Hypotheses

Null Hypothesis (H0): The true mean weight is 10 grams. (μ=10)
Alternative Hypothesis (Ha): The true mean weight is not 10 grams. (μ $\neq =$ 10)

Step 2: Calculate the t-statistic

The formula for the t-statistic is:

$t = \frac{x ˉ - μ}{s / n}$

Where:

$\overset{x}{ˉ}$ = sample mean = 9.8 grams
μ = hypothesized population mean = 10 grams
s = sample standard deviation = 0.5 grams
n = sample size = 15

$t = \frac{9.8 - 10}{\frac{0.5}{15}}$

$t = \frac{- 0.2}{\frac{0.5}{3.873}}$

$t = \frac{- 0.2}{0.129}$

$t \approx - 1.55$

Step 3: Determine the Degrees of Freedom

The degrees of freedom (df) for a one-sample t-test is calculated as:

$df = n - 1$

$df = 15 - 1 = 14$

Step 4: Use the T.DIST Function

The engineer wants to find the probability of observing a t-statistic of -1.55 or less. This is a left-tailed probability, which can be calculated directly using the T.DIST function.

T.DIST(x, deg_freedom, cumulative)

x: -1.55 (the calculated t-statistic)
deg_freedom: 14
cumulative: TRUE (since we want the cumulative probability, or the area to the left of our t-statistic)

The function call would be: T.DIST(-1.55, 14, TRUE)

Results Table

	Parameter	Value
	A	B
1	x (t-statistic)	-1.55
2	deg_freedom	14
3	cumulative	TRUE
4	T.DIST result	0.071723565

Interpretation

The T.DIST function returns a value of approximately 0.0717. This means that, assuming the true mean weight is 10 grams, there is a 7.17% chance of observing a sample mean of 9.8 grams or less from a sample of 15 parts.

Since this probability (p-value) is greater than the common significance level of 0.05, the engineer would fail to reject the null hypothesis. There is not enough statistical evidence to conclude that the new manufacturing process produces parts with a mean weight different from 10 grams.