Beware!
split(/\s=\s/) will not do what you think it should.

Try this one:

#!/usr/bin/perl
use strict;
use warnings;

my %hash;
open(ANSWERTALLY, "answertally.dat");

while(<ANSWERTALLY>) {
  next unless /^Running totals:/ .. /^Averages:/; # by davorg
  next if /^$/; # next line if empty
  next if /^Running totals:/; # by davorg
  last if /^Averages:/; # by davorg
  my ($key, $value) = split(/:/); # you actually just want to split by
+ ':'
  $value =~ s/\s*//g; # remove whitespace globally
  $hash{$key} = $value unless $value eq ""; # add to hash only if valu
+e is holding data
}
close(ANSWERTALLY);

for my $key(sort keys %hash) {
  print "\$key: $key\t\$hash{\$key}: $hash{$key}\n";
}

__END__
__OUTPUT__
$key: Question 1        $hash{$key}: 60
$key: Question 2        $hash{$key}: 123
$key: Question 3        $hash{$key}: 99
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