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

Using Excel functions like t.inv.2t, t.inv, t.dist, t.dist.rt, and t.dist.2t, students can calculate critical values and probabilities for the Student’s t-distribution, essential for constructing confidence intervals and hypothesis testing. Degrees of freedom, calculated as $n-1$, and alpha levels guide these calculations. Understanding left-tail, right-tail, and two-tail probabilities enables precise interpretation of t-scores, enhancing skills in statistical inference, confidence intervals, and p-value determination within the framework of sampling distributions 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, the probability input corresponds to 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 working 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 a right tail probability, 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.

These Excel functions leverage the properties of the t-distribution, which is characterized by its degrees of freedom and is used when the population standard deviation is unknown and the sample size is relatively small. The critical values obtained are essential for constructing confidence intervals using the formula:

\[\bar{x} \pm t_{\alpha/2} \times \frac{s}{\sqrt{n}}\]

where \(\bar{x}\) is the sample mean, \(s\) is the sample standard deviation, and \(n\) is the sample size.

Mastering the use of Excel’s T.INV.2T and T.INV functions enhances your ability to accurately and efficiently find critical t-values for various statistical analyses, whether dealing with two-tailed confidence intervals or one-tailed hypothesis tests.

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) $(-1.293,1.293)$ (B) $1.373$ (C) $-1.9925$
• B. (A) $(-1.293,1.293)$ (B) $-1.373$ (C) $1.9925$
• C. (A) $(-1.666,1.666)$ (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, \(df = n - 1\).

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 less than the negative of a value or greater than 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 negative inputs will cause errors.

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 approximately 0.09, or a 9% chance of a t score being either below -1.71 or above 1.71.

Understanding these Excel functions and their inputs allows for efficient calculation of probabilities associated with t scores, which is fundamental in hypothesis testing and confidence interval estimation in statistics.

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 idea is to calculate the left-tail probabilities (cumulative probabilities) for each t score and then subtract the smaller from the larger. Specifically, the probability that the t score lies between -1.76 and 2.3 is the difference between the cumulative probability at 2.3 and the cumulative probability at -1.76.

Using the t-distribution cumulative function, denoted as \(F(t, \nu)\) where \(t\) is the t score and \(\nu\) is the degrees of freedom, the probability between the two t scores is:

\[P(-1.76 < T < 2.3) = F(2.3, 85) - F(-1.76, 85)\]

For example, the cumulative probability for \(t = 2.3\) with 85 degrees of freedom is approximately 0.988, and for \(t = -1.76\) it is approximately 0.041. Subtracting these gives:

\[0.988 - 0.041 = 0.947\]

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

This method mirrors the approach used for the normal distribution, where the probability between two z-scores is found by subtracting the cumulative probability of the lower z-score from that of the higher z-score. The t-distribution function, often available in statistical software or spreadsheet programs as T.DIST with a cumulative option, facilitates this calculation by directly providing the left-tail 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(probability, degrees_freedom) function. Here, $\mathrm{probability}$ represents the combined alpha level in both tails, calculated as $\mathrm{\alpha \; =\; 1\; -\; ext\{confidence\; level\}}$. The $\mathrm{degreesextunderscore\; freedom}$ is found by subtracting one from the sample size, $\mathrm{df\; =\; n\; -\; 1}$. For example, for a 95% confidence interval with a sample size of 61, alpha is 0.05 and degrees of freedom is 60. Enter =T.INV.2T(0.05, 60) to get the positive critical value. The negative critical value is simply the negative of this result. This method allows you to find critical values for any confidence level and sample size, which is more flexible than using traditional t-tables.

What Excel function do I use to find the left tail probability for a given t score?

To find the left tail probability (the probability that a t score is less than a given value) in Excel, use the =T.DIST(x, degrees_freedom, TRUE) function. Here, $x$ is the t score, $\mathrm{degreesextunderscore\; freedom}$ is $\mathrm{n\; -\; 1}$, and the last argument TRUE specifies that you want the cumulative probability up to $x$. For example, if you have a t score of 1.71 and 60 degrees of freedom, enter =T.DIST(1.71, 60, TRUE) to get the left tail probability, which tells you the chance of observing a t score less than 1.71.

How can I calculate the right tail probability for a t score in Excel?

To calculate the right tail probability (the probability that a t score is greater than a given value) in Excel, use the =T.DIST.RT(x, degrees_freedom) function. Here, $x$ is the t score and $\mathrm{degreesextunderscore\; freedom}$ is $\mathrm{n\; -\; 1}$. For example, if your t score is 1.71 with 60 degrees of freedom, enter =T.DIST.RT(1.71, 60) to find the probability of getting a t score greater than 1.71. This function directly returns the right tail probability without needing to use the complement rule.

What is the difference between the Excel functions T.INV and T.INV.2T?

The Excel function T.INV(probability, degrees_freedom) returns the t score corresponding to a left tail probability, meaning it finds the t value where the cumulative probability to the left equals the given probability. In contrast, T.INV.2T(probability, degrees_freedom) returns the positive t critical value for a two-tailed test, where $\mathrm{probability}$ is the combined alpha level in both tails. For example, for a 95% confidence interval, T.INV.2T(0.05, df) gives the critical value for both tails combined, while T.INV(0.025, df) gives the critical value for the left tail only (half of alpha). This distinction is important when constructing confidence intervals or performing hypothesis tests.

How do I find two-tail probabilities from t scores using Excel?

To find two-tail probabilities from t scores in Excel, use the =T.DIST.2T(x, degrees_freedom) function. Here, $x$ is the positive t score (always enter the positive value), and $\mathrm{degreesextunderscore\; freedom}$ is $\mathrm{n\; -\; 1}$. This function calculates the combined probability of getting a t score less than $-x$ or greater than $x$. For example, if your t score is ±1.71 with 60 degrees of freedom, enter =T.DIST.2T(1.71, 60) to find the probability of observing a t score outside this range, which is useful for two-tailed hypothesis tests.

How do I calculate degrees of freedom for t-distribution problems in Excel?

Degrees of freedom (df) for t-distribution problems are calculated as the sample size minus one, $\mathrm{df\; =\; n\; -\; 1}$. In Excel, you can compute this by typing =n - 1 where n is your sample size. For example, if your sample size is 61, enter =61 - 1 to get 60 degrees of freedom. This value is essential as it is used as an input in all t-distribution functions like T.INV.2T, T.DIST, and others to accurately calculate critical values and probabilities.