Question

A reduction in heart rate variability is associated with elevated levels of stress, since the body continues to pump adrenaline after high-stress situations, even when at rest. Suppose the standard deviation of heartbeats in the general population is 20 beats per minute, and that the population of heart rates is normal. A random sample of 50 individuals leading high-stress lives has a sample variance of 200 beats per minutes. If appropriate, test using level of significance α = 0.05 whether the population standard deviation for those leading high-stress lives is lower than that in the general population.

Answer 1

H0: sigma = 20
Ha: sigma < 20

here sample std. dev. s = sqrt(200) = 14.14
n = 50

Test statistic,
chi-square = (n-1)*(s/sigma)^2
= 49 * 200/20^2
= 24.5

p-value = P(chi-square < 24.5)
p-value = 0.0013

critical value of chi-square = 33.93
Reject H0 if chi-square < 33.93

As p-value < 0.05, reject H0
There is sufficient evidence to conclude that the population standard deviation for those leading high-stress lives is lower than that in the general population.