Finding Probabilities and T Critical Values - Excel: Videos & Practice Problems

Using Excel functions like =T.INV.2T(alpha, degrees_freedom) helps find t critical values for confidence intervals, where alpha is the combined tail probability and degrees of freedom equals sample size minus one. For left or right tail probabilities, =T.INV(probability, degrees_freedom) is used, converting right tail probabilities via the complement rule. To find probabilities from t scores, functions such as =T.DIST(x, degrees_freedom, TRUE) for left tail, =T.DIST.RT(x, degrees_freedom) for right tail, and =T.DIST.2T(x, degrees_freedom) for two-tail probabilities are applied, facilitating inferential statistics and hypothesis testing.

Finding t Critical Values From Probabilities - Excel

Finding t critical values is essential for constructing confidence intervals and conducting hypothesis tests involving the t distribution. Using Excel simplifies this process, allowing you to calculate critical values for any confidence level and sample size, which is more flexible than relying on traditional t tables.

To find the critical value for a two-tailed confidence interval in Excel, use the function =T.INV.2T(probability, degrees_freedom). Here, probability represents the combined area in both tails, denoted as α, which equals 1 minus the confidence level. The degrees of freedom is calculated as n - 1, where n is the sample size. For example, for a 95% confidence interval with a sample size of 61, α = 0.05 and degrees of freedom = 60. The function returns the positive critical value \(t_{\alpha/2}\), and the negative critical value is simply its negative counterpart, \(-t_{\alpha/2}\).

When dealing with one-tailed probabilities, Excel uses the function =T.INV(probability, degrees_freedom). For a left-tail probability, input the given probability directly. For instance, if the left-tail probability is 0.24 with a sample size of 40, degrees of freedom is 39, and the function returns the t score corresponding to that cumulative probability.

For right-tail probabilities, since Excel’s T.INV function requires a left-tail probability, convert the right-tail probability using the complement rule: left-tail probability = 1 − right-tail probability. For example, if the right-tail probability is 0.39 with 39 degrees of freedom, inputting 1 - 0.39 = 0.61 into the function yields the corresponding t score.

Understanding these Excel functions enhances your ability to accurately find critical t values for various statistical analyses. This knowledge supports constructing precise confidence intervals and performing hypothesis tests, especially when sample sizes are small and the t distribution is more appropriate than the normal distribution.

For a sample size of 75, find the following t-scores.
(A) The critical values needed to create a $90\%$ confidence interval.
(B) The t-score with a left probability of $0.087$.
(C) The t-score with a right tail probability of $0.025$.

• A. (A) $\left(-1.293,1.293\right)$ (B) $1.373$ (C) $-1.9925$
• B. (A) $\left(-1.293,1.293\right)$ (B) $-1.373$ (C) $1.9925$
• C. (A) $\left(-1.666,1.666\right)$ (B) $-1.373$ (C) $1.9925$
• D. (A) $\left(-1.666,1.666\right)$ (B) $1.373$ (C) $-1.9925$

Answer: (C)

Finding Probabilities From t-Scores

When working with the t distribution in Excel to find probabilities from t scores, it is essential to select the correct function based on the type of probability you want to calculate. For left tail probabilities, which represent the chance of obtaining a t score less than a specific value, the =T.DIST function is used. This function requires three inputs: the t score (x), the degrees of freedom (df), and a logical value indicating whether to calculate the cumulative distribution (always TRUE for cumulative probabilities). The degrees of freedom are typically calculated as the sample size minus one, i.e., \(df = n - 1\).

For right tail probabilities, which represent the chance of obtaining a t score greater than a certain value, the =T.DIST.RT function is appropriate. This function only requires two inputs: the t score and the degrees of freedom. Unlike =T.DIST, it does not have a cumulative argument, and entering one will result in an error.

When calculating two-tail probabilities, which involve the probability of a t score being either below the negative of a value or above the positive of that value, the =T.DIST.2T function is used. This function also takes only two inputs: the positive t score and the degrees of freedom. It is important to input the positive value of the t score, as using a negative value will cause an error.

For example, consider a sample of 61 kittens where the t score calculated is 1.71. The degrees of freedom would be \$61 - 1 = 60$. Using =T.DIST(1.71, 60, TRUE) yields approximately 0.95, indicating a 95% probability of obtaining a t score less than 1.71. Conversely, =T.DIST.RT(1.71, 60) gives about 0.05, representing a 5% chance of a t score greater than 1.71. For the two-tail probability, =T.DIST.2T(1.71, 60) results in roughly 0.09, or a 9% chance of a t score being either below -1.71 or above 1.71.

Understanding these functions and their inputs allows for accurate calculation of probabilities associated with t scores in Excel, facilitating statistical analysis involving the t distribution.

Find the following probabilities for a sample size of 75:
(A) _____________
(B) $P(T>1.65)=$ _____________
(C) OR $T>1.65){=}_{}$ ____________

• A. (A) $0.0516$ (B) $0.0516$ (C) $0.1032$
• B. (A) $0.9484$ (B) $0.9484$ (C) $0.0516$
• C. (A) $0.0516$ (B) $0.0516$ (C) $0.0516$
• D. (A) $0.9484$ (B) $0.9484$ (C) $0.8968$

Answer: (A)

Finding Probabilities From t-Scores Example 1

To find the probability of obtaining a t score between two values, such as between -1.76 and 2.3, for a sample size of 86, you can use the cumulative distribution function (CDF) of the t-distribution. The degrees of freedom for the t-distribution is calculated as the sample size minus one, so here it is 85.

The key approach involves calculating the left-tail probabilities (cumulative probabilities) for each t score and then subtracting the smaller left-tail probability from the larger one. This method is similar to finding the probability between two z-scores in the normal distribution.

First, calculate the left-tail probability for the larger t score, 2.3, which represents the probability that the t score is less than 2.3. Then, calculate the left-tail probability for the smaller t score, -1.76, which is the probability that the t score is less than -1.76. The probability of the t score falling between -1.76 and 2.3 is the difference between these two cumulative probabilities.

Mathematically, this can be expressed as:

\[P(-1.76 < T < 2.3) = P(T < 2.3) - P(T < -1.76)\]

Using statistical software or functions such as T.DIST in Excel, you input the t score, degrees of freedom (85), and specify that you want the cumulative probability (left-tail). For example, =T.DIST(2.3, 85, TRUE) returns approximately 0.988, and =T.DIST(-1.76, 85, TRUE) returns approximately 0.041.

Subtracting these gives:

\[0.988 - 0.041 = 0.947\]

This result means there is about a 94.7% probability that a t score from this distribution falls between -1.76 and 2.3.

This method highlights how cumulative probabilities for the t-distribution can be used to find probabilities over intervals, reinforcing understanding of degrees of freedom, cumulative distribution functions, and the relationship between t scores and probabilities.

Here’s what students ask on this topic:

How do you find t critical values for confidence intervals using Excel?

To find t critical values for confidence intervals in Excel, use the =T.INV.2T function. This function requires two inputs: the combined tail probability (alpha) and the degrees of freedom (df). Alpha is calculated as $\alpha =1-\mathrm{confidence\; level}$. Degrees of freedom are found by subtracting one from the sample size, $\mathrm{df}=n-1$. For example, for a 95% confidence level and sample size of 61, alpha is 0.05 and df is 60. Enter =T.INV.2T(0.05, 60) to get the positive critical value. To find the negative critical value, simply add a negative sign before the result. This method allows you to find critical values for any confidence level and sample size, which is more flexible than using traditional tables.

What Excel functions are used to find probabilities from t scores?

Excel provides three main functions to find probabilities from t scores: =T.DIST, =T.DIST.RT, and =T.DIST.2T. Use =T.DIST to find the left tail probability (probability of a t score less than a given value). It requires three inputs: the t score ($x$), degrees of freedom ($df$), and true for cumulative probability. For right tail probabilities (t score greater than a value), use =T.DIST.RT, which takes the t score and degrees of freedom as inputs. For two-tail probabilities (probability of t score being less than negative value or greater than positive value), use =T.DIST.2T with the positive t score and degrees of freedom. These functions help convert t scores into probabilities essential for hypothesis testing and confidence intervals.

How do you calculate degrees of freedom for t distribution in Excel?

Degrees of freedom (df) for the t distribution are calculated as the sample size minus one: $\mathrm{df}=n-1$. In Excel, you can calculate this by typing =n - 1 where n is your sample size. For example, if your sample size is 40, enter =40 - 1 to get 39 degrees of freedom. This value is essential as it is used as an input in Excel functions like =T.INV.2T, =T.DIST, and others to find critical values and probabilities based on the t distribution.

How do you find the t score for a given left tail probability in Excel?

To find the t score corresponding to a left tail probability in Excel, use the =T.INV function. This function takes two inputs: the left tail probability and the degrees of freedom. For example, if the left tail probability is 0.24 and the sample size is 40, first calculate degrees of freedom as $df\; =\; 40\; -\; 1\; =\; 39$. Then enter =T.INV(0.24, 39) to get the t score. This t score represents the value below which 24% of the t distribution lies. This method is useful for hypothesis testing and determining critical values for one-tailed tests.

How can you find the t score for a right tail probability using Excel?

Excel's =T.INV function requires a left tail probability as input, so to find a t score for a right tail probability, you must convert it to a left tail probability using the complement rule: $\mathrm{left\; tail\; probability}=\; 1\; -\mathrm{right\; tail\; probability}$. For example, if the right tail probability is 0.39 and the sample size is 40, calculate degrees of freedom as $df\; =\; 40\; -\; 1\; =\; 39$. Then enter =T.INV(1 - 0.39, 39) in Excel. This returns the t score such that the area to the right of this value is 0.39. This approach is essential for finding critical values in right-tailed hypothesis tests.

What is the difference between =T.DIST, =T.DIST.RT, and =T.DIST.2T functions in Excel?

The Excel functions =T.DIST, =T.DIST.RT, and =T.DIST.2T calculate probabilities from t scores but differ in the tail probabilities they return. =T.DIST returns the cumulative left tail probability (probability that the t score is less than a given value) and requires three inputs: t score, degrees of freedom, and true for cumulative. =T.DIST.RT returns the right tail probability (probability that the t score is greater than a given value) and takes two inputs: t score and degrees of freedom. =T.DIST.2T returns the two-tail probability (probability that the t score is either less than the negative or greater than the positive of a given value) and also takes two inputs: positive t score and degrees of freedom. Understanding these differences helps in selecting the correct function for hypothesis testing and confidence interval calculations.