Testing for unknown proportion

[EvLer]

Homework Statement

for testing Ho : p = 0.2 vs H1: p > 0.2 when will the power of Rao score test be larger:
1. alpha = 0.05 n = 10
2. alpha = 0.01 n = 10

Homework Equations

Rao score test: reject Ho when |sqrt(n)*(p' - po)/sqrt(po (1-po)| > z(alpha)

po in this case = 0.2 and p' is estimated proportion from the given data.
z comes from N(0,1)

power is defined as 1 - Beta, where Beta is the prob. of making type II error in testing hypothesis, i.e. accepting Ho when H1 is true

The Attempt at a Solution

I think (2) since with higher alpha it is easier to reject Ho, but I am a bit doubtful on this.
Thanks.

 

[mighty2000]

I would like to clarify that the power of a test is not affected by the significance level (alpha). Power is determined by the sample size (n), effect size, and the level of significance (alpha) chosen. Therefore, the power of the Rao score test will be the same in both scenarios.

In this case, the power of the test will increase with a larger sample size (n). This is because with a larger sample size, the estimated proportion (p') will be closer to the true proportion (po), leading to a larger value for the test statistic and a higher chance of rejecting the null hypothesis (Ho).

Additionally, the power of the test will also increase with a larger effect size, which is the difference between the true proportion (po) and the null hypothesis proportion (p). This means that if the true proportion is significantly different from the null hypothesis proportion, the test will be more likely to detect this difference and reject the null hypothesis.

In summary, the power of the Rao score test will be larger with a larger sample size (n) and a larger effect size, regardless of the chosen significance level (alpha). It is important to carefully consider these factors when designing a hypothesis test to ensure an appropriate level of power.