Platelet rich plasma (PRP) therapy is used to accelerate the healing of injured tendons, ligaments and joints, using plasma derived from the patient's own blood. A study involving 255 randomly selected patients with shoulder pain caused by rotator cuff tears reported that after receiving a PRP injection, 225 patients experienced at least a 50% reduction in pain.

Is there evidence that PRP therapy relieves pain in more than 85% of patients with such shoulder pain?

a) Write appropriate hypotheses.

b) Check the assumptions and conditions.

c) Calculate the test statistic and find the P-value.

d) State your conclusion in plain English. Use a significance level of 5%.

e) Calculate a 90% confidence interval for the true proportion of patients with shoulder pain caused by rotator cuff tears whose pain is reduced by the PRP therapy, and interpret your interval in context.

f) Explain how the confidence interval supports your conclusion from part d).