Question is in the description.

Question

A new COVID-19 vaccine is developed and the developer company claims that the vaccine’s protection

lasts longer than their competitors. In an experiment with 55 subjects, the company showed that their

vaccine protects for 6.2 months with a standard deviation of 1.5 months. Note that, when tested on an

independent set of 55 patients, the old vaccine developed by their competitors protect for 5.8 months with

a standard deviation of 1.1 months. With 5% level of significance, can we state that the new vaccine really

protects for a longer duration? (Please state H0 and HA)

Answer 1

for μ₁ = mean protection of old≥ agent μ₁ = mean protection of new agent

Agents were tested on the same population against the same vector (virus)

H₀ : μ₁ ≥ μ₂

H_a : μ_{1 <} μ₂

Based on problem description a one sided, two sample T test, no interaction, pooled population variance is appropriate.

s²_p = (s²₁(n₁-1) + s²₂(n₂-1) / (n₁ + n₂ - 2)

note:x^- = x bar = mu hat = sample mean

t = ( x^-₁ - x^-₂)/√ (s²_p(1/n₁+1/n₂))

p-value equals 6.57099e-141, test statistic T equals -199.985230

With 5% level of significance, we can state that the new agent really

'protects' for a longer duration based on unstated definition of protection.