Question

a researcher wants to investigate whether there is a difference in the aerobic capacity between two different exercise programs. group 1 participates in a running program, and group 2 participates in a cycling program. after 8 weeks, the aerobic capacity is measured for both groups. below are the sample data:
group 1 (running program): sample size ($n_1$) = 3, sample mean ($overline{x}_1$) = 40, sample variance ($s_1^2$) = 4
group 2 (cycling program): sample size ($n_2$) = 4, sample mean ($overline{x}_2$) = 45, sample variance ($s_2^2$) = 9.
question 1:
what is the t - test statistic to test the difference between the groups?
-2.48
-4.35
2.48
4.35

Explanation:

Step1: Calculate the pooled variance

The formula for pooled variance \(s_p^2=\frac{(n_1 - 1)s_1^2+(n_2 - 1)s_2^2}{n_1 + n_2-2}\)
Substitute \(n_1 = 3\), \(s_1^2=4\), \(n_2 = 4\), \(s_2^2 = 9\)
\(s_p^2=\frac{(3 - 1)\times4+(4 - 1)\times9}{3 + 4-2}=\frac{2\times4+3\times9}{5}=\frac{8 + 27}{5}=\frac{35}{5}=7\)

Step2: Calculate the t - statistic

The formula for t - statistic \(t=\frac{\bar{X}_1-\bar{X}_2}{\sqrt{s_p^2(\frac{1}{n_1}+\frac{1}{n_2})}}\)
Substitute \(\bar{X}_1 = 40\), \(\bar{X}_2 = 45\), \(s_p^2=7\), \(n_1 = 3\), \(n_2 = 4\)
\(t=\frac{40 - 45}{\sqrt{7(\frac{1}{3}+\frac{1}{4})}}=\frac{- 5}{\sqrt{7\times\frac{4 + 3}{12}}}=\frac{-5}{\sqrt{7\times\frac{7}{12}}}=\frac{-5}{\sqrt{\frac{49}{12}}}=\frac{-5}{\frac{7}{\sqrt{12}}}\)
\(t=\frac{-5\sqrt{12}}{7}\approx\frac{-5\times3.464}{7}\approx - 2.48\)

Answer:

-2.48