code: def test_value_counts_with_normalize(dtype):
      if getattr(dtype, "storage", "") == "pyarrow":
          exp_dtype = "double[pyarrow]"
      elif getattr(dtype, "storage", "") == "pyarrow_numpy":
          exp_dtype = np.float64
      else:
          exp_dtype = "Float64"
      ser = pd.Series(["a", "b", "a", pd.NA], dtype=dtype)
      result = ser.value_counts(normalize=True)
      expected = (
          pd.Series(
              [2, 1], index=ser[:2], dtype=exp_dtype, name="proportion"
          )
          / 3
      )
      tm.assert_series_equal(result, expected)
  
imports: import operator
  import numpy
  import pytest
  from pandas.compat.pyarrow import pa_version_under12p0
  from pandas.core.dtypes.common import is_dtype_equal
  import pandas
  import pandas._testing
  from pandas.core.arrays.string_arrow import ArrowStringArray, ArrowStringArrayNumpySemantics
label_desc: Use image node_id 35 for calling a global function with example usage: test_value_counts_with_normalize(dtype) without return types