Question

An article in Medicine and Science in Sports and Exercise ["Maximal Leg-Strength Training Improves Cycling Economy in Previously Untrained Men" (2005, Vol. 37, pp. 131–136)] studied cycling performance before and after 8 weeks of leg-strength training. Seven previously untrained males performed leg-strength training 3 days per week for 8 weeks (with four sets of five replications at 85% of one repetition maximum). Peak power during incremental cycling increased to a mean of 315 watts with a standard deviation of 16 watts. Construct a 95% confidence interval for the mean peak power after training.

Answer 1

Answer:

[300.202 , 329.798]

Step-by-step explanation:

The 95% confidence interval is given by the interval

$\large [\bar x-t^*\frac{s}{\sqrt n}, \bar x+t^*\frac{s}{\sqrt n}]$

where

$\large \bar x$ is the sample mean  

s is the sample standard deviation  

n is the sample size (n = 7)  

$\large t^*$ is the 0.05 (5%) upper critical value for the Student's t-distribution with 6 degrees of freedom (sample size -1), which is an approximation to the Normal distribution for small samples (n<30).

Either by using a table or the computer, we find  

$\large t^*= 2.447$

and our 95% confidence interval is

$\large [315-2.447*\frac{16}{\sqrt{7}}, 315+2.447*\frac{16}{\sqrt{7}}]=\boxed{[300.202,329.798]}$

Answer 2

Using the t-distribution, it is found that the 95% confidence interval for the mean peak power after training is (301.2, 328.8).

We have the standard deviation for the sample, thus, the t-distribution is used.

• The sample mean is $\overline{x} = 315$.
• The sample standard deviation is $s = 16$.
• The sample size is $n = 7$.

The interval is:

$\overline{x} \pm t\frac{s}{\sqrt{n}}$

To build the interval, we need to find the critical value, which looking at the t-table for a 95% confidence interval with 8 - 1 = 7 df is t = 2.4469.

Then, replacing the values:

$315 - 2.4469\frac{16}{\sqrt{8}} = 301.2$

$315 + 2.4469\frac{16}{\sqrt{8}} = 328.8$

The 95% confidence interval for the mean peak power after training is (301.2, 328.8).

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