Question

Rescue operations conducted by firefighters involve intense physical activity associated with high level of cardiovascular stress. In 2020, a group French researchers investigated the claim that a combination of deep breathing and mental imagery promotes cardiovascular recovery in firefighters. The researchers recruited a random sample of 21 male professional firefighters who were medically fit with five years of firefighting experience. To get a baseline measurement, the researchers subjected the firefighters to the Cooper test, a popular maximal test of aerobic fitness during which a participant runs as far as possible in 12 minutes. The mean distance among the 21 participants in the baseline measurements was 2.681 km with a standard deviation of 0.352 km. If the researchers wanted to estimate the mean distance firefighters can run in 12 minutes with 90% confidence and a margin of error of no more than 0.1 km, how large a sample size would they need? Use a t* value rounded to three decimal places and the sample information above to calculate your answer. Your answer should be an integer value.

Answer 1

To estimate the required sample size, we can use the formula:

n = (t* * σ / E)²

Where:
n = sample size
t* = critical value (obtained from the t-distribution table for the desired confidence level)
σ = standard deviation of the population
E = maximum margin of error

Given information:
Mean distance (μ) = 2.681 km
Standard deviation (σ) = 0.352 km
Margin of error (E) = 0.1 km
Confidence level = 90%

First, we need to find the critical value (t*) corresponding to a 90% confidence level. Since the sample size is small (less than 30), we should use the t-distribution instead of the normal distribution.

The degrees of freedom (df) for this calculation would be n - 1. However, since we don't know the sample size yet, we'll use the conservative approach and assume the worst-case scenario, which is the smallest possible sample size (n = 1). Therefore, the degrees of freedom would be 1 - 1 = 0.

Using a t-distribution table or a statistical software, we find that the critical value for a 90% confidence level and df = 0 is approximately 6.314.

Now, we can substitute the values into the formula:

n = (t* * σ / E)²
n = (6.314 * 0.352 / 0.1)²
n = 22.316²
n ≈ 498

Therefore, the researchers would need a sample size of at least 498 firefighters to estimate the mean distance firefighters can run in 12 minutes with a 90% confidence level and a margin of error no more than 0.1 km.