Independent Samples T Test: Hypothesis Testing and Interpretation

10 Questions

When can we reject the null hypothesis in a t-test?

When p-value is less than or equal to the significance level

What does a p-value larger than the significance level indicate in a t-test?

Insufficient evidence to reject null hypothesis

What does it mean when we fail to reject the null hypothesis in a t-test?

Evidence against alternative hypothesis

How can we compare the mean of a single sample to a known value?

Both A and B

In a one-sample t test with 78 observations, what degrees of freedom are used to determine critical t value?

77 degrees of freedom

What is the purpose of the independent samples t test?

To determine if there is a significant difference between the means of two groups

In hypothesis testing of the independent samples t test, what does the null hypothesis typically assume?

The sample means come from populations with equal means

What assumption must be satisfied for an independent samples t test to be valid?

The variances of the two groups must be equal

When should the Welch's correction be applied in the context of an independent samples t test?

When the variances of the two populations are unequal

What is the primary objective of conducting an independent samples t test?

To establish if there is a statistically significant difference in means between two groups

Study Notes

Independent Samples T Test

The independent samples t test is a method used to compare the means of two different groups with independent observations. It is often used when there are only two conditions being compared, such as before and after treatment, or control vs experimental group. This statistical analysis helps researchers determine if there is a significant difference between the means of these two groups.

Hypothesis Testing

Hypothesis testing is the process of using data and statistical methods to make decisions based on the uncertainty inherent in experimental outcomes. In the context of the independent samples t test, we typically have our null hypothesis (for example, assuming that the sample means come from populations with equal means) and a corresponding alternative hypothesis (for example, assuming that the population means differ by some amount).

Assumptions

To apply an independent samples t test, certain assumptions must be met:

Observations are independent, meaning that each observation occurs randomly and is unrelated to other observations.
Both populations have normal distributions. If this assumption does not hold, or it is unknown whether one population has a skewed distribution, the Welch's correction can be applied.
The variances of the populations are equal. If they are unequal, the Welch's correction can also be applied.

Interpretation of Results

If the p-value obtained from the t-test is less than or equal to the chosen significance level, we reject the null hypothesis and conclude that there is significantly different in means between the two samples. However, if the p-value is larger than the significance level, we fail to reject the null hypothesis. Such a failure to reject the null hypothesis does not prove the null hypothesis to be true; rather, it provides evidence against the alternative hypothesis.

Mean Comparison

In order to perform a one-sample t test for comparing the mean of a single sample to a known value, we can either use the standard formula for calculating the t statistic or use a table or software to find the critical t value based on the desired confidence level. For example, given a sample size of 78, a sample mean of 9, a known expected value of 5, and a confidence level of 95%, we would first calculate the t statistic and then compare it to the critical t value for 77 degrees of freedom (which corresponds to n = 78 - 1) and a confidence level of 95%.