Step 1: Read both statements carefully.
Statement I claims modern Homo sapiens arose in Australia, and Statement II gives the time frame of about $75000$ to $10000$ years ago.
Step 2: Recall where modern humans originated.
Fossil and genetic evidence shows modern Homo sapiens arose in Africa, then migrated across the continents into Asia, Europe, Australia and the Americas.
Step 3: Judge Statement I.
Since the true place of origin is Africa, not Australia, Statement I is incorrect.
Step 4: Recall the time of origin.
Modern humans appeared during the last ice age, dated to roughly $75000$ to $10000$ years ago.
Step 5: Judge Statement II.
This time frame matches the textbook value, so Statement II is correct.
Step 6: Combine the verdicts.
Statement I is wrong and Statement II is right, which is option 1.
\[ \boxed{\text{Statement I is incorrect but Statement II is correct}} \]