You should first initialize you SQLiteDatabase, so convert this:
SQLiteDatabase database;
database.delete(ActivityTableDBOpenHelper.ACTIVITY_TABLE,null,null);
to this:
SQLiteDatabase database = new SQLiteDatabase(this); // or dbHelper.getWritableDatabase(); if you have a dbHelper
database.delete(ActivityTableDBOpenHelper.ACTIVITY_TABLE,null,null);
Else the database object is null and you get the error.
Here you could find a whole example for all operations http://www.vogella.com/tutorials/AndroidSQLite/article.html
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solved How to delete all Table records in Sqlite?