SQL & PL/SQL :: Compile A Procedure Within Another Procedure?

I wanted to Compile a Procedure within another procedure.

Step 1:

CREATE OR replace PROCEDURE Compile_test
IS
var1 NUMBER(20);
BEGIN
SELECT user_id
INTO var1
FROM dummy;
dbms_output.Put_line('the output is '
||var1);
END;

Step 2:

CREATE OR replace PROCEDURE Compile_test_in (proc_name VARCHAR2)
IS
var2 VARCHAR2(20);
BEGIN
var2 := 'proc_name';
EXECUTE IMMEDIATE ('alter procedure ||var2|| compile');
END;

Step 3:
exec compile_test_in ('compile_test')

When trying the step 3, i am getting the below error message.

Error at line 1
ORA-04050: invalid or missing procedure, function, or package name
ORA-06512: at "MUTHU.COMPILE_TEST_IN", line 6
ORA-06512: at line 1