Question

Assume the samples are random samples from distributions that are reasonably normally distributed, and that a t-statistic will be used for inference about the difference in sample means. State the degrees of freedom used. Find the proportion in a t-distribution above 2.1 if the samples have sizes \(n_{1}=12\) and \(n_{2}=12\).

Short answer

Given the t-value of 2.1, the degrees of freedom are 22 (which were calculated from \( n_{1} = 12 \), \( n_{2} = 12 \) samples). However, the exact proportion in the t-distribution above 2.1 would depend on the use of a statistical calculator or statistical software for accurate results.

Step-by-step solution

Calculate Degrees of Freedom

The degrees of freedom are calculated as the sum of the sizes of the two groups, minus 2. So the degrees of freedom used will be \( (n_{1} + n_{2}) - 2 \) where \( n_{1} = 12 \) and \( n_{2} = 12 \). This results in 22 degrees of freedom.

Understand the distribution of t-values

The t-statistic follows a t-distribution. It's a type of probability distribution that is symmetric and bell-shaped, similar to the standard normal distribution but it's affected by the sample size through the degrees of freedom. Typically, the larger the degrees of freedom, the closer the t-distribution is to the standard normal distribution.

Find the proportion

To find the proportion in the t-distribution above the t-statistic of 2.1, one should refer to a t-distribution table for 22 degrees of freedom. However, since actual values may vary based on the specific t-table used, this step requires using a statistical calculator or statistical software to find the precise p-value for a t-statistic of 2.1 with 22 degrees of freedom. Please note that the exact proportion might vary based on the utilized software or methods, and this step generally will be dependent on the actual application.

Key concepts

Degrees of Freedom

Understanding the concept of degrees of freedom (DF) is fundamental in statistics, particularly when dealing with inferences about sample means using t-distributions. In the context of our exercise, degrees of freedom refer to the number of independent values in a dataset that are free to vary when some statistics are being computed. The term originates from the concept that certain restrictions set limits on data values.

For example, when computing the sample variance, one degree of freedom is lost because the mean's value influences the variance calculation. In the case of two independent samples, the degrees of freedom are calculated as the total number of observations in both samples minus the number of studied groups. Following our exercise, we calculated the degrees of freedom as 22 by adding the two sample sizes and subtracting two \( (n_{1} + n_{2}) - 2 = (12 + 12) - 2 = 22 \).

In the realm of hypothesis testing, specifically when using the t-distribution to estimate the mean of a normally distributed population when the sample size is small, knowing the correct degrees of freedom is crucial. This is because the shape of the t-distribution and thus the critical values of the t-statistic depend on the degrees of freedom.

T-Statistic

The t-statistic is a pivotal tool in inferential statistics and is used when the population standard deviation is unknown and the sample size is relatively small. It is akin to the z-score of the normal distribution, but takes into account the extra uncertainty introduced by estimating the population standard deviation using the sample data.

In hypothesis testing, the t-statistic is used to determine how far away your sample mean is from the null hypothesis mean as a multiple of the standard error. It's calculated by the formula \( t = \frac{\bar{x} - \mu_{0}}{\frac{s}{\sqrt{n}}} \) where \( \bar{x} \) is the sample mean, \( \mu_{0} \) is the null hypothesis mean, \( s \) is the sample standard deviation, and \( n \) is the sample size. As seen in our exercise, we're concerned with the t-distribution of the statistic 2.1, given the degrees of freedom we've established. Identifying the proportion of values that fall above this t-statistic is crucial in determining p-values for hypothesis tests.

Sample Means Inference

Inferring about the sample means stands at the core of statistical analysis, letting us draw conclusions about the population from which the sample is taken. The process involves constructing confidence intervals or conducting hypothesis tests based on the sample statistics.

The inference of sample means typically involves assuming that the samples come from a normally distributed population or that they are large enough for the Central Limit Theorem to apply, which says that the sampling distribution of the sample mean will be approximately normal regardless of the population distribution, given a sufficient sample size. In hypothesis testing, particular for small samples from approximately normal distributions, the t-distribution provides a more appropriate framework for inference than the normal distribution because it accounts for the variability introduced by estimating population parameters from sample statistics.

In our textbook example, to infer the difference in sample means using a t-statistic, we take into account the degrees of freedom, which reflects both sample sizes, to correctly gauge the variability expected in our t-distribution and thereby determine the statistical significance of our observed difference.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by a symmetric, bell-shaped curve. It's also known as the Gaussian distribution and is a cornerstone of many statistical procedures because it aptly models many natural phenomena.

In the context of hypothesis testing and confidence interval estimation, the normal distribution is most relevant because many sample statistics are normally distributed or approximately so, especially with larger sample sizes. This distribution is defined by its mean and standard deviation, which determine the distribution's location and width, respectively. The rule of thumb is that approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

However, with smaller samples, as noted in the exercise and previous sections, the t-distribution provides a better model because it has heavier tails than the normal distribution, allowing for the increased variability that smaller sample sizes bring. As the sample sizes increase and the degrees of freedom grow, the t-distribution approaches the normal distribution, which is why for large sample sizes, the normal distribution is often used in inference.