A test was made of $H _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ versus $H _ { 1 } : \mu _ { 1 }

Question

A test was made of $H _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ versus $H _ { 1 } : \mu _ { 1 } < \mu _ { 2 }$. The sample means were $\bar { x } _ { 1 } = 13$ and $\bar { x } _ { 2 } = 12$, the sample standard deviations were $s _ { 1 } = 4$ and $s _ { 2 } = 3$, and the sample sizes were $n _ { 1 } = 16$ and $n _ { 2 } = 15$.
How many degrees of freedom are there for the test statistic?
A) 15
B) 16
C) 11
D) 14

Quizplus · Accepted Answer

The degrees of freedom for the test statistic in a two-sample t-test, where the variances are assumed to be unequal (Welch's approximation), can be calculated using the formula: $$df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1-1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2-1}}$$ Substituting the given values: $$df = \frac{\left(\frac{4^2}{16} + \frac{3^2}{15}\right)^2}{\frac{\left(\frac{4^2}{16}\right)^2}{16-1} + \frac{\left(\frac{3^2}{15}\right)^2}{15-1}} \approx 29.07$$ Since degrees of freedom must be an integer, and given the options, the closest and correct approximation is not directly listed in the options provided, indicating a potential misunderstanding in the question's setup or in the interpretation of the options. However, the correct approach to calculate degrees of freedom for the test statistic in such a context involves using the formula provided, which typically results in a non-integer that is then rounded. Given the options and the typical rounding rule, none directly match the calculated value, suggesting a reevaluation of the provided options or the calculation method for this specific context.

A Test Was Made Of H0:μ1=μ2H _ { 0 } : \mu _ { 1 } = \mu _ { 2 }H0​:μ1​=μ2​

A Test Was Made Of $H _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$