Questions: When n=19, the probability P(t>1.18)=[?] Round to the nearest ten-thousandth.

Transcript text: When $\mathrm{n}=19$, the probability \[ \mathrm{P}(\mathrm{t}>1.18)=[?] \] Round to the nearest ten-thousandth.

Solution

Solution Steps

To find the probability $ \mathrm{P}(\mathrm{t}>1.18) $ when $ \mathrm{n}=19 $, we need to use the t-distribution. The degrees of freedom for the t-distribution is $ \mathrm{n} - 1 $. We will use a statistical library in Python to calculate the cumulative distribution function (CDF) for the t-distribution and then subtract it from 1 to find the probability that $ \mathrm{t} > 1.18 $.

Step 1: Determine Degrees of Freedom

Given $ n = 19 $, the degrees of freedom for the t-distribution is calculated as: \[ \text{df} = n - 1 = 19 - 1 = 18 \]

Step 2: Calculate Cumulative Probability

The cumulative probability for $ t \leq 1.18 $ with 18 degrees of freedom is: \[ \text{CDF}(1.18, 18) \approx 0.8733 \]

Step 3: Calculate Probability for $ t > 1.18 $

The probability that $ t > 1.18 $ is the complement of the cumulative probability: \[ P(t > 1.18) = 1 - \text{CDF}(1.18, 18) \approx 1 - 0.8733 = 0.1267 \]