T.DIST.RT

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

Syntax:

T.DIST.RT(x, deg_freedom)

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.

Example:

If x contains 2.5 and deg_freedom contains 19:

T.DIST.RT(2.5, 19)

returns 0.010870206

x:

deg_freedom:

Result:

0.010870206

Application:

Scenario: A coffee shop owner is concerned that their espresso machines are not consistently dispensing the correct amount of coffee. The target fill volume is 45 milliliters (ml) per shot. A random sample of 12 shots is taken, and the volumes are recorded. The owner wants to know if the machines are over-filling the shots, and they want to test this at a 5% significance level.

Hypothesis Testing:

Null Hypothesis (H0): The average coffee volume is equal to 45 ml (μ=45).
Alternative Hypothesis (Ha): The average coffee volume is greater than 45 ml (μ>45).

This is a one-tailed (right-tailed) test because the owner is only concerned with the machines over-filling the shots.

Sample Data and Calculations:

The following table shows the sample data and the steps for calculating the t-statistic:

	Shot Number	Volume (ml)
	A	B
1	1	45.8
2	2	44.5
3	3	46.1
4	4	45.2
5	5	45.5
6	6	46
7	7	44.9
8	8	45.7
9	9	46.3
10	10	45.4
11	11	45.6
12	12	45.9

From this data, we can calculate the following:

Sample size (n): 12
Sample mean ( $\overset{x}{ˉ}$ ): 45.575 ml
Sample standard deviation (s): 0.546 ml
Degrees of freedom (df): n−1=12−1=11

Now, we calculate the t-statistic using the formula:

$t = \frac{x ˉ - μ}{s / n}$

$t = \frac{45.575 - 45}{0.546/ 12}$

$t = \frac{0.575}{0.546/3.464}$

$t = \frac{0.575}{0.158}$

$t \approx 3.639$

Using the T.DIST.RT Function:

To find the probability (p-value) of getting a t-statistic of 3.639 or higher, we would use the T.DIST.RT function with our calculated values.

	Parameter	Value
	A	B
1	x (t-statistic)	3.639
2	deg_freedom	11
3	Function	T.DIST.RT(3.639, 11)
4	Result (p-value)	0.0019

Interpreting the Result:

The p-value is 0.0019 (or 0.19%).
The significance level (α) is 5% (or 0.05).
Since the p-value (0.0019) is less than the significance level (0.05), we reject the null hypothesis.

Conclusion:

Based on the sample data, there is strong statistical evidence to conclude that the espresso machines are, on average, over-filling the coffee shots. The owner should recalibrate their machines to ensure they are dispensing the correct volume.