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 less than -1.4 if the samples have sizes \(n_{1}=30\) and \(n_{2}=40\)

Short answer

The degrees of freedom for this test is 68. To find the proportion of the t-distribution that is less than -1.4, use a t-distribution table or statistical software with the degrees of freedom and the t-value. The result is the probability that the difference between the two sample means is less than -1.4.

Step-by-step solution

Calculate the Degrees of Freedom

The degrees of freedom for a two-sample t-distribution is calculated using the formula: \( df = n_1 + n_2 - 2 \). Here, \( n_1 \) is the size of the first sample and \( n_2 \) is the size of the second sample. In this case, \( n_1 = 30 \) and \( n_2 = 40 \). Substituting these values into the formula gives: \( df = 30 + 40 - 2 = 68 \). So, the degrees of freedom for this test is 68.

Find the Proportion

To find the proportion of the t-distribution that is less than -1.4, one can use a t-distribution table or statistical software with the degrees of freedom and the t-value. From the t-distribution table or statistical software, you can find the proportion which is less than -1.4.

Interpret the Result

The result you get in step 2 is the probability that the difference between the two sample means is less than -1.4. It's important to note that t-distribution proportions are often given as absolute values, so a t-value of -1.4 will give the same proportion as a t-value of 1.4. The sign only indicates which side of the distribution you're on.

Key concepts

Degrees of Freedom

Understanding the concept of 'degrees of freedom' is key in analyzing data sets and interpreting statistical results. In statistics, 'degrees of freedom' refers to the number of independent values or quantities which can be assigned to a statistical distribution. In the context of a two-sample t-test, the degrees of freedom is calculated based on the sample sizes of the groups being compared. Specifically, the formula is df = n_1 + n_2 - 2, where n_1 and n_2 are the sample sizes of the first and second groups, respectively. The subtraction of two accounts for the estimation of two sample means.
In the provided exercise, the degrees of freedom for the two-sample t-test with sample sizes of 30 and 40 is 68. This value is crucial for determining the correct distribution to use when looking up the proportion of the t-statistic, as tables and software rely on it. The higher the degrees of freedom, the closer the t-distribution approaches the normal distribution.

Two-Sample T-Test

The two-sample t-test is a statistical method used to determine whether there is a significant difference between the means of two independent samples. This test is particularly useful when comparing samples of different sizes or when data does not adhere strictly to a normal distribution, but is reasonably normal. For example, the exercise you've provided involves using a t-test to compare the sample means from two different groups.
The two-sample t-test assesses whether the population means are distinct from each other by considering the variance within each of the samples. The calculated t-statistic is then compared to a critical value from the t-distribution with the appropriate degrees of freedom to determine if the difference in means is statistically significant.

Normal Distribution

The normal distribution, often known as a bell curve, is a fundamental concept in statistics. This continuous probability distribution is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. When sample sizes are large, the Central Limit Theorem dictates that the sampling distribution of the sample mean will tend to be normal, regardless of the shape of the population distribution.
This is relevant because t-tests, including the two-sample t-tests from our exercise, are based on the presumption that the distribution of the sample means will be normal or near-normal - especially as the sample sizes become large (typically greater than 30). This allows us to make inferences about the population from which the samples are taken, assuming normality.

Inferential Statistics

Inferential statistics are techniques that allow us to use samples to make generalizations about populations. This branch of statistics takes data from a sample and makes inferences about the larger population from which the sample was drawn. In the exercise context, the t-statistic derived from the two-sample t-test is an example of inferential statistics. It helps us determine the probability of observing a sample mean difference at least as extreme as the one calculated if there were no actual difference in the population means (the null hypothesis).
Through inferential statistics, we can estimate population parameters, test hypotheses, and make predictions. It's all about making the best conclusion based on available data, acknowledging that a certain level of uncertainty is always present.

Sample Mean Comparison

Sample mean comparison is a crucial process in statistical analysis, where we compare the means from two different samples to draw conclusions about their populations. This can be done using a two-sample t-test, as in our exercise. The primary goal of this test is to determine whether the two sample means are statistically significantly different from one another. This may involve testing a null hypothesis, which often posits that there is no difference between the two population means (that any observed difference is due to sampling variation).
The t-test gives us a t-value, which, compared with a critical value from the t-distribution - factoring in the degrees of freedom - helps us to make this determination. A low probability result (p-value) indicates that the observed sample mean difference is unlikely to have occurred by chance, thereby leading us to reject the null hypothesis and accept that the population means are different.