Question given to me:
A drug to reduce blood pressure is administered to n = 300 patients. After 15 minutes, 276 had blood pressure in the normal range.The company wants to claim that the drug is effective for more than 90% of patients. Using a significance level of $\alpha = .05$, determine if they can do this, or if they should make the more conservative claim that the drug is at least 90% effective.
My instructor's "correct" hypotheses:
$campaign 1: H_{0}: \widehat{p}= .9, H_{a}: \widehat{p}> .9, campaign 2: H_{0}: \widehat{p}= .9, H_{a}: \widehat{p}< .9$
My main question is in the title; wouldn't "at least" just be inclusive of 90% and since it is continuous, be essentially the same as "more than"?
Yes. I give a calculus-based explanation.
The (Riemann) integral doesn't care if you remove one point. In fact, it doesn't care if you remove countably many points. The integral will still be the same.
Now in these tests you use the normal distribution, a continuous distribution. The area is computed using an integral, approximated by your tables. It doesn't matter if you have the strict inequality or not, the result will still be the same.