1.6 Comparing two independent means | Inferential Statistics | Comparing two groups | UvA

The video introduces comparing two independent groups on a quantitative variable using a t-test for two independent means. It also covers the assumption of equality of population variances and calculating a confidence interval.

A t-test for two independent means is used with a quantitative response variable and a binary independent variable distinguishing two independent samples. Examples include comparing television watch time between employed and unemployed individuals or happiness scores between parents and non-parents.

Key assumptions include independent samples and normally distributed samples (though the t-test is robust for large samples). Statistical hypotheses are expressed as the difference between population means (null hypothesis: difference is zero; alternative hypothesis: difference is not zero, or greater/smaller than zero).

The test statistic (T) is calculated as the difference in sample means minus the expected value under the null hypothesis (zero), divided by the standard error. The standard error is derived from the square root of the sum of group variances divided by their sample sizes. The test statistic follows a Student's t-distribution with a complex degrees of freedom formula.

An example tests if a raw meat diet is healthier for cats than canned food. Cats are randomly assigned to diets, and health scores are measured. After checking for normality, a one-sided alternative hypothesis (raw diet leads to higher health scores) is tested. The calculated t-value is 1.87 with 274.95 degrees of freedom, resulting in a P-value of 0.032, which is smaller than the significance level of 0.05, leading to the rejection of the null hypothesis.

An alternative t-test assumes equal population variances. This simplifies the calculation of degrees of freedom (total sample size minus 2) and standard error (pooled standard deviation times the square root of the sum of reciprocals of sample sizes). While this can lead to larger degrees of freedom and a potentially smaller standard error, the unequal variances version is generally recommended for safety. The rule of thumb for making the extra assumption is when standard deviations differ by less than a factor of two.

The confidence interval is calculated as the difference in sample means plus and minus t-values (associated with the required confidence level and degrees of freedom) multiplied by the standard error. For the cat diet example, a 95% confidence interval for the mean difference is -0.01 to 0.54. Since zero (no difference) lies within this interval, a two-sided test would have been non-significant, indicating that the one-sided test was more conservative.