Question

Pilates is a popular set of exercises for the treatment of individuals with lower back pain. The method has six basic principles: centering, concentration, control, precision, flow, and breathing. An article reported on an experiment involving 84 subjects with nonspecific low back pain. The participants were randomly divided into two groups of equal size. The first group received just educational materials, whereas the second group participated in 6 weeks of Pilates exercises. The sample mean level of pain (on a scale from 0 to 10) for the control group at a 6-week follow-up was 5.2 and the sample mean for the treatment group was 3.2; both sample standard deviations were 2.3.

a. Does it appear that true average pain level for the control condition exceeds that for the treatment condition? Carry out a test of hypotheses using a significance level of .01 (the cited article reported statistical significance at this α, and a sample mean difference of 2.1 also suggests practical significance).
b. Does it appear that true average pain level for the control condition exceeds that for the treatment condition by more than 1? Carry out a test of appropriate hypotheses.

Answer 1

a) We reject H₀ We find difference between the mean of the two groups

b) we accept H₀ : There is no statistical difference ( in the scale of pain) when the difference in samples means is 1

Explanation:

Group 1: Only with educational material

Sample size n₁ = 42

Sample mean x₁ = 3,2

Sample standard deviation s₁ = 2,3

Group 2: With exercises with the treatment of individuals

Sample size n₂ = 42

Sample mean x₂ = 5,2

Sample standard deviation s₂ = 2,3

Test Hypothesis

Null Hypothesis H₀ x₂ - x₁ = 0 or x₁ = x₂

Alternative Hypothesis Hₐ x₂ - x ₁ > 0 or x₂ > x₁

Significance level α = 0,01

Sample sizes are n₁ = n₂ > 30

We use z-test

for α = 0,01 z(c) ≈ 2,32

To calculate z(s)

z(s) = ( x₂ - x₁ ) / √ (2,3)²/42 + (2,3)²/42

z(s) = ( 5,2 - 3,2 )/ √0,2519

z(s) = 2 / 0,5

z(s) = 4

Comparing z(s) and z(c)

z(s) > z(c)

z(s) is in the rejection region. We reject H₀ the true average pain level for the control condition exceeds that for the treatment condition.

b) Does the true average pain level for the control condition exceed that for the treatment condition by more than 1.

In this case

z( s) = x₂ - x₁ / 0,5

z(s) = 1 /0,5

z(s) = 2

Comparing z(s) and z(c) now

z(s) < z(c) 2 < 2,32

And z(s) is in the acceptance region ( for the same significance level) and we should accept H₀ equivalent to say that the two groups statistical have the same mean