hypothesis testing question?

Question

"The U.S. Postal Service advertises that at least 63.4% of all third class mail is read by the recipients. An environmental group, concerned about waste production wants to test this assertion. They take a sample of 220 households and find that only 58.7% read their third class mail."

I thought the answer would be, H0 : P ≥ 0.634 and H1 : P < 0.634 as it says "at least 63.4%" in the question
but the answer is "H0 : P = 0.634 and H1 : P < 0.634" and I don't understand why

Answer 1

Because if the real population proportion is GREATER than advertised proportion, then all is well and good.

 

The concern here is that the proportion is LESS than what is advertised. Specifically, is there statistical

evidence to show that the real population proportion does NOT meet the percent that is advertised.

 

The sample proportion is p=58.7% = 0.587

 

1-p = 1 - 0.587 =    0.413

 

The standard deviation is  sqrt( 0.587*0.413/220) = 0.0332....

 

Now we need a confidence constant k. Let's pick 99%.

So the confidence coefficient is k=2.576 for 99% confidence.

 

 

The difference is 0.587 - 0.634 =  -0.047

The test statistic is -0.047/0.0332 =-1.415662651.... > -1.65 >-2.576

 

This is within the threshold for even 90% confidence, so the sample supports the claim