Question

We wish to validate the suspicion that the duration of the antibodies generated by exposure to COVID 19 has an average of not more than 180 days. 50 patients are monitored and the following result is obtained. the result of the experiment shows that X = 184 days; s = 12.03 days.

a) clearly establishes the hypotheses.

b) define the areas of acceptance and rejection.

c) Carry out the analysis by hand and express your conclusion in a complete sentence.

d) Calculate the p-value.

e) What is the probability of making a type 1 error in this test?

f) Calculate the probability of making a type 2 error if in reality: u = 175 days, u = 182 days

Answer 1

(a) The hypothesis being tested is:

H₀: µ 180

H_a: µ > 180

(b) The curve is:

(c) The test statistic, t = (x - µ)/s/√n

t = (184 - 180)/12.03/√50

t = 2.351

(d) The p-value is 0.0114.

Since the p-value (0.0114) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that µ > 180.

(e) Probability = 0.05

(f)

(i) u = 175 days:

Sample Size, n = 50

Type I error rate, α = 0.05

True mean, μ = 175

Null Hypothesis mean, μ0 = 180

Standard Deviation, σ = 12.03

Type II error = 1 - [Φ(|μ−μ0|/σ/√n - z1−α)] = 1 - [|175 - 180|/12.03/√50 - 1.96] = 1 - [-2.94 - 1.96]

= 0.0976 [From the z-table]

(ii) u = 182 days

Sample Size, n = 50

Type I error rate, α = 0.05

True mean, μ = 182

Null Hypothesis mean, μ0 = 180

Standard Deviation, σ = 12.03

Type II error = 1 - [Φ(|μ−μ0|/σ/√n - z1−α)] = 1 - [|182 - 180|/12.03/√50 - 1.96] = 1 - [-1.18 - 1.96]

= 0.6801 [From the z-table]