Question

Let \(T\) have a \(t\) -distribution with 10 degrees of freedom. Find \(P(|T|>2.228)\) from either Table III or by using \(\mathrm{R}\).

Short answer

The probability that a random variable \(T\) with a t-distribution of 10 degrees of freedom has an absolute value greater than 2.228 could be determined by using the t-distribution table or the statistical software R. Please note that the exact value may vary depending on the table used or computation precision, but it gives a good estimate for solving this type of exercises.

Step-by-step solution

- Identify relevant information

Here, the degrees of freedom is identified as \(df=10\). The value of interest (the point at which the cumulative probability should be determined) is identified as \(t=2.228\). Our goal is to find the probability where the absolute value of \(T\) is greater than \(2.228\).

- Use the t-distribution table

In the t-distribution table (Table III), the degrees of freedom are listed on the left, while the t-values are on the top. First, locate the row for 10 degrees of freedom. Next, find the value closest to 2.228 (if it's not exactly on the table, use the closest one). That value gives the probability that \(T ≤ 2.228\). However since we want to find \(P(|T|>2.228)\), and knowing that the t-distribution is symmetric around zero, you would subtract this probability value from 1 and multiply it by 2 to get the two-tailed probability.

– Calculate with R programming language

Alternatively, this can be calculated directly using R, a programming language for statistical computing. In R, the function pt() is used to compute the t-distribution. However, as the function gives the probability of values less or equal to the given point, we would specify lower tail as false to get the probability for values greater than the given point. For the two-tailed test, this value would then be doubled. So, the required code would be: \(2*(1 - pt(2.228, df = 10)) \).

Key concepts

Understanding Statistical Probability

Statistical probability is a core concept in statistics that measures the likelihood of a specific event occurring. In the case of the textbook exercise, we're dealing with a probability distribution known as the t-distribution, which is particularly useful when dealing with smaller sample sizes or when the population standard deviation is unknown.
When calculating the probability of a t-score, or any statistical measure, we are often interested in the area under the curve of the probability distribution function. For the exercise mentioned, the goal is to find the probability that the absolute value of T is greater than 2.228, denoted as P(|T| > 2.228). This probability includes both tails of a symmetric t-distribution since the t-value is positive on the right tail and its negative counterpart on the left tail. This is why, in Step 2 of the solution, the two-tailed probability is obtained by subtracting the table value from 1 and then doubling the result.
Overall, understanding statistical probability allows researchers and students alike to make informed conclusions about data, test hypotheses, and predict future trends by quantifying uncertainty.

Degrees of Freedom Explained

Degrees of freedom (df) are a vital concept in statistics, representing the number of independent values that can vary in an analysis without breaking any constraints. In the context of the t-distribution, degrees of freedom are equivalent to the sample size minus one (n-1). The solution’s reference to '10 degrees of freedom' means that the data set in question would have 11 observations.
Understanding degrees of freedom is essential when dealing with statistical distributions. It affects the shape of the distribution curve; for the t-distribution, as the degrees of freedom increase, the curve becomes more like the standard normal distribution. In the textbook solution, the degrees of freedom help us find the exact row in the statistical table that corresponds to our data set. They also play a critical role in the calculation performed in R, as seen in Step 3, where df = 10 is specified as an argument in the function call. This critical value helps to determine the precise probability that we are seeking for a given statistical test.

R Programming for Statistics

R is an open-source programming language that has become a powerhouse for statistical analysis, data visualization, and data science. R provides a vast collection of packages and functions designed specifically for statistical calculations. One of these functions is pt(), used for calculating the cumulative probability for a given t-value with a specified number of degrees of freedom.
In the exercise solution, R succinctly performs the calculation that would otherwise require manual lookup in statistical tables. By inputting the t-value and degrees of freedom into the function and setting the lower tail argument to false, we're asking R to compute the probability of obtaining a t-score greater than the specified value. This greatly simplifies the process for students and researchers and allows for quick and accurate probability calculations. For those learning statistics, mastering R can be an invaluable tool, not only to check work against traditional methods but also to perform complex data analyses that are impractical to do by hand.

Code Snippet Overview

2 * (1 - pt(2.228, df = 10)) is the R code mentioned in the solution. This line doubles the probability of getting a t-value greater than 2.228 with 10 degrees of freedom, hence giving us the desired two-tailed test probability.