I would like to know who the people of Shevet Ephraim were (they were
descendants of Joseph), and how it came they came to be that they made erroneous
calculations, and as a result, left Egypt 30 years before the appointed time, and
what happened to them when they did? I understand that out of 1,000 of
them, only ten survived their premature exit from Egypt.

The main problem is that there is no hard historical evidence for the
Exodus as described in the Bible. So according to the facts currently
known, one has to assume that the entire story around those events is
just that - a story. It may well be constructed around certain real
events, but as long as the Exodus has no place in history, the events
directly connected with it must be considered just as fictitious. It
would not be possible to analyze events that have most likely never
taken place the way described in the Old Testament.

Scriptor

IF the Bible is the word of God, and given that He is perfect in
everything, then He does not lie and therefore the bible is all true.
Including the Exodus. This does take faith. God chose the
Hebrews/Jews/Israelites as His "chosen" people, got them out of Egypt
and gave them a set of laws. They stayed together as a people right up
till the time Jesus arrived and set up the " new covenant". All this
is set out in the Bible. And the fact that the Bible itself has
survived for so long. God did promise that He would preserve it.
Revelation 20;5 states.. Write, because these words are faithful and
true. Where certain people went and why is only of passing interest
but it is very important that people do not miss the main theme of the
Bible, to change their behaviour to please God. Not just the Jews, but
all people. The reminder is at Second Peter 3;9-13...He is patient
with you because He does not desire any to be destroyed but desires
ALL to attain to repentance...

Not looking for archeological records - just Midrashic sources, etc. -
giving more details on what happened, the background of how a group of
people made an error that got most of them killed, on how acting
prematurely can be disastrous.

If one wanted more background on the side-issue of: Did Exodus occur,
and I thoroughly believe it did, see: The Miracles of Exodus  A
scientist's discovery of the extraordinary natural caseos of the
Biblical stories -- by Colin J Humphreys.