Should I add additional unit tests for this helper method I've used that simply helps refactoring?

I have a method:

public static void AddPersonToPeopleList(List<PersonModel> people, PersonModel person)
        {
            CheckWhiteSpace(person.FirstName, "FirstName");
            CheckWhiteSpace(person.LastName, "LastName");

            people.Add(person);
        }

That calls the helper methdod:

public static void CheckWhiteSpace(string name, string paramName) {
            if (string.IsNullOrWhiteSpace(name)) {
                throw new ArgumentException("You passed in an invalid parameter", paramName);
            }
        }

Which simply checks for null or white space in my string with the parameter name, and throws an exception if it does.

        [Theory]
        [InlineData("Mick", "", "LastName")]
        [InlineData("", "Adams", "FirstName")]
        public void AddPersonToPeopleList_ShouldFail(string firstName, string lastName, string param)
        {
            PersonModel newPerson = new PersonModel { FirstName = firstName, LastName = lastName };
            List<PersonModel> people = new List<PersonModel>();

            Assert.Throws<ArgumentException>(param, () => DataAccess.AddPersonToPeopleList(people, newPerson));
        }

My test is designed to simply make sure an exception is thrown if either the first name or second name is empty.

My question is whether I need to also make a test for the helper method I used (CheckWhiteSpace) or whether the test I've made takes care of testing the functionality of that helper method and my original method?