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 5:
what is the standard error of the difference between two means?
1.15
1.50
2.02
3

Explanation:

Step1: Recall the formula for standard error of the difference between two means

When the variances are unequal (we use the formula \(SE=\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}\))
Here, \(n_{1} = 3\), \(s_{1}^{2}=4\), \(n_{2}=4\), \(s_{2}^{2}=9\)

Step2: Substitute the values into the formula

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Wait, let's re - check. If we assume equal variances (pooled variance formula \(s_{p}^{2}=\frac{(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2}\), \(SE = s_{p}\sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}\))

\(s_{p}^{2}=\frac{(3 - 1)\times4+(4 - 1)\times9}{3 + 4-2}=\frac{8 + 27}{5}=\frac{35}{5} = 7\)

\(SE=\sqrt{7\times(\frac{1}{3}+\frac{1}{4})}=\sqrt{7\times\frac{4 + 3}{12}}=\sqrt{7\times\frac{7}{12}}=\sqrt{\frac{49}{12}}\approx2.02\)

Answer:

2.02