Question

1. (15 points) ) A scientific journal decides to publish only results that are "statistically at the o 0.05 level." As a model of this situation, assume all statistical tests are of the form H0:0-00 vs. HA: θ -9A (where θο represents the standard and θΑ represents the new theory). A result is published if and only if Ho is rejected, with a P- value less than 0.05. For every test, Pr(reject Hol%) Let po denote the proportion of all experiments in which Ho is correct and let pa denote the proportion of all experiments in which HA is correct (so po+ PA Pr(reject Holed 0.05. Let π 1). (a) Of the articles published, what proportion have correct results? That is, find Pr(0-0Alreject Ho). (Note: many people naively believe that the journal is guaranteeing that 95% of published articles are correct.) 0.75 and pA-o2. Evaluate your answer to part (a) for this special b ) Suppose that case. c) Suppose pA = 0.2 but that π is unknown. Maximize the quantity in part (a) with respect to π. That is, find the value of π that maximizes Pro-θΑΙ reject Ho).

Answer 1

a) P(Reject$H_0$)=P(Reject$H_0$$/\theta_0$)*P($\theta_0$)+P(Reject$H_0$$/\theta_A$)*P($\theta_A$​​​​​​​)

=0.05*p₀+$\pi$p_A

P($\theta=\theta_A$/Reject$H_0$)= P(Reject$H_0$​​​​​​​$/\theta_A$)*P($\theta_A$​​​​​​​) /P(Reject$H_0$)

=$\pi$p_A/(0.05*p₀+$\pi$p_A)

b) P($\theta=\theta_A$/Reject$H_0$) =$\pi$p_A/(0.05*p₀+$\pi$p_A)=0.75*0.2/(0.05*0.8+0.75*0.2)=0.7895

c) If $\pi$ is unknown then P($\theta=\theta_A$/Reject$H_0$) =$\pi$*0.2_/(0.05*0.8+0.2$\pi$)

  

0.2

The above function is maximum at the maximum value of $\pi$​​​​​​​ which is 1