First let me say that I appreciate your thoughtful skepticism. "Preaching to the choir" is always dangerous and answering your questions and challenges is an excellent reality check.
As I see it, there are four possible explanations for Randolph's report:
1) he made it up himself;
2) he heard a hoax broadcast;
3) he misunderstood a news broadcast about the Earhart situation; and
4) he actually heard a transmission from Amelia.
I agree.
Your argument against the first possibility is that the local newspaper had not yet published a story about the disappearance so Randolph could not have known about it and so could not have made up this story. HELLO!, the kid had a RADIO and could have learned about the Earhart situation before the story was published in his local newspaper.
I did not say that Randolph was not aware of Earhart's disappearance. He certainly WAS aware that Earhart was missing. That's probably why, when he first heard her (or what he believed was her), he hollered to his father , "Hey Paw! I got Miss Earhart!" who "came running in" from the kitchen. What I said was "printed news of post-loss signals had not yet reached Rock Springs." Both Randolph and the local newspaper editor were unaware that other post-loss signals had been heard. The story in the local paper was headlined "First Radio Contact With Miss Earhart Made By Rock Springs Boy."
It is, of course, possible that 16 year-old Dana Randolph made the whole thing up but his father Cyrus also claimed to have heard the transmissions so if it was all a hoax, Cyrus Randolph was in on it. According to the newspaper story, "Dana's uncle, Victor Randolph, who lives next door, came in and was told about the reception of the call for help. "Everybody wants to know about that,' he told his nephew. 'Get down town and report that.' Cyrus and Victor Randolph immediately went to the police station to learn where the report should be made. They were directed to the local Department of Commerce radio operator" who, in turn notified Washington.
The Randolph family was African-American. To suggest that they were perpetrating a hoax is to suggest that an African-American family in Rock Springs, Wyoming in 1937 made up a story about hearing distress calls from a famous white woman and reported the fable, not to the newspaper, but to the police department and then to a representative of the federal government. You can make your own judgement about the likelihood of that happening.
As for the second possibility, anybody who had also heard of the situation could have made up the story and a hoax broadcast.
It is true that anyone can make up a story. Making a hoax broadcast that only one person hears is trickier. I don't know how you do that. Using the Ionospheric Communications Enhanced Profile Analysis and Circuit Prediction Program (ICEPAC), Bob Brandenburg has calculated the chances of Dana Randolph hearing Earhart on the 5th harmonic of of 3105 Khz as 0.016. It turns out that at the same time Dana Randolph was hearing what he thought was Amelia Earhart, Mrs. Crabb (we don't have her first name) in Toronto was hearing fragments of a conversation between a man and woman she believed to be Amelia Earhart and Fred Noonan. Mrs. Crabb heard what she heard on 18639 Khz, the 6th harmonic of 3105 KHz - a different frequency than Randolph but also a harmonic of Earhart's frequency. The probability of Crabb hearing Earhart was 0.009. In both cases - Randolph and Crabb - hearing Earhart would be an unusual, but far from impossible event and entirely consistent with the intermittent, fading in and out signal they reported.
In our evaluation of all of the known reported post-loss signals we identified 28 apparent hoaxes but in all cases it appears that the reporter was the hoaxer - in other words, claiming to hear something they did not really hear. We found no evidence of anyone sending a bogus transmission.
As to being on a reef southeast of Howland, any hoaxter could look at any map of the Pacific or a globe and notice that the closest islands are southeast of Howland, so it would not be a great leap for a hoaxter to put that language in his fake message and to also use the common word "ship" for the plane on the reef. And this other hoaxter didn't need to read about it in Randolph's local newspaper.
All true. Let's look at what it would take to perpetrate a hoax transmission. First you must have a transmitter or access to a transmitter. Then you have to decide what frequency to send your hoax transmission on. If you choose a commercial broadcast frequency lots of people are going to hear your hoax - but that didn't happen. If you choose a harmonic of Earhart's frequency that would make your hoax more credible, but nobody was listening for Earhart on harmonics of her frequency so you can't count on anyone hearing your hoax. What fun is that?
In the third scenario, the kid heard part of a shortwave broadcast in the standard 19 meter international broadcast band, 15,100 to 15,800 kcs: .....static (fade) "......the ship may be afloat but some believe (suddenly louder) the ship is on a reef southeast of Howland Island (fade) but the Coast Guard is still....." Anybody who has ever listened on a shortwave radio to international broadcasts knows that these signals fade in an out so this scenario is realistic.
We, of course, don't know everything that was being said in news broadcasts about the search, but what we do know is that in 1937 a female newscaster, or "presenter" as our British cousins say, was extremely unusual if not unheard of. We presume that Dana and his father were aware that Miss Earhart was a woman. We also know that at the time Dana Randolph said he heard Amelia say "ship on reef" (Sunday morning, July 4) there was no speculation yet in the print news media that the plane might be on land.
And the fourth possibility is that he heard a legitimate Earhart message.
After carefully considering the alternatives we feel that this is the most likely explanation and we have judged this report of a post-loss message to be credible. That doesn't mean that we can prove that this particular message was sent by Earhart. It just means that, based on what we know about it, the message appears to be credible.