diff --git a/qiskit/primitives/estimatorv2converter.py b/qiskit/primitives/estimatorv2converter.py
index 73c6a15f8c3b..f803ad55e9e9 100644
--- a/qiskit/primitives/estimatorv2converter.py
+++ b/qiskit/primitives/estimatorv2converter.py
@@ -15,7 +15,8 @@
 """
 from __future__ import annotations
 
-from collections.abc import Iterable, Sequence
+from collections.abc import Iterable
+from warnings import warn
 
 import numpy as np
 
@@ -46,7 +47,7 @@ def __init__(
 
     def run(
         self, pubs: Iterable[EstimatorPubLike], *, precision: float | None = None
-    ) -> BasePrimitiveJob[PrimitiveResult[PubResult]]:
+    ) -> PrimitiveJob[PrimitiveResult[PubResult]]:
         coerced_pubs = [EstimatorPub.coerce(pub, precision) for pub in pubs]
 
         job = PrimitiveJob(self._run, coerced_pubs)
@@ -60,18 +61,19 @@ def _run_pub(self, pub: EstimatorPub) -> PubResult:
         circuit = pub.circuit
         observables = pub.observables
         parameter_values = pub.parameter_values
-        precision = pub.precision
+        if pub.precision is not None:
+            warn("Precision is not defined yet. Ignored now.")
 
         out_shape = np.broadcast_shapes(pub.observables.shape, pub.parameter_values.shape)
-        param_broad = np.broadcast_to(pub.parameter_values, out_shape)
-        param_object = np.zeros(pub.parameter_values.shape, dtype=object)
-        param_array = pub.parameter_values.as_array(circuit.parameters)
-        for idx in np.ndindex(pub.parameter_values.shape):
+
+        param_object = np.zeros(parameter_values.shape, dtype=object)
+        param_array = parameter_values.as_array(circuit.parameters)
+        for idx in np.ndindex(parameter_values.shape):
             param_object[idx] = param_array[idx].tolist()
 
         obs_list = []
         param_list = []
-        for obs, param in np.broadcast(pub.observables, param_object):
+        for obs, param in np.broadcast(observables, param_object):
             spo = SparsePauliOp.from_list(list(obs.items()))
             obs_list.append(spo)
             param_list.append(param)
@@ -80,8 +82,9 @@ def _run_pub(self, pub: EstimatorPub) -> PubResult:
         result = self._estimatorv1.run([circuit] * size, obs_list, param_list).result()
         evs = result.values.reshape(out_shape)
         stds_list = [
-            np.sqrt(dat.get("variance", 0) / dat.get("shots", 1)) for dat in result.metadata
+            np.sqrt(dat.get("variance", np.nan) / dat.get("shots", 1)) for dat in result.metadata
         ]
         stds = np.array(stds_list).reshape(out_shape)
         data_bin = self._make_data_bin(pub)(evs=evs, stds=stds)
-        return PubResult(data_bin, metadata={"precision": precision})
+        metadata = result.metadata
+        return PubResult(data_bin, metadata=metadata)
diff --git a/test/python/primitives/test_estimatorv2converter.py b/test/python/primitives/test_estimatorv2converter.py
index dfea663c54b1..48425edb3244 100644
--- a/test/python/primitives/test_estimatorv2converter.py
+++ b/test/python/primitives/test_estimatorv2converter.py
@@ -19,7 +19,7 @@
 
 from qiskit.circuit import Parameter, QuantumCircuit
 from qiskit.circuit.library import RealAmplitudes
-from qiskit.primitives import Estimator, EstimatorV2Converter, StatevectorEstimator
+from qiskit.primitives import Estimator, EstimatorV2Converter
 from qiskit.primitives.containers.bindings_array import BindingsArray
 from qiskit.primitives.containers.estimator_pub import EstimatorPub
 from qiskit.primitives.containers.observables_array import ObservablesArray
@@ -133,7 +133,6 @@ def test_run_single_circuit_observable(self):
                 self.subTest(f"{val}")
                 result = est.run([(qc, op, val)]).result()
                 np.testing.assert_allclose(result[0].data.evs, target)
-                self.assertEqual(result[0].metadata["precision"], 0)
 
         with self.subTest("One parameter"):
             param = Parameter("x")
@@ -267,21 +266,24 @@ def test_run_numpy_params(self):
     def test_precision_seed(self):
         """Test for precision and seed"""
         precision = 0.1
-        shots = int(1 / precision)  # TODO: define the relation between precision and shots
-        estimator_v1 = Estimator({"seed": 1, "shots": shots})
+        shots = int(1 / precision**2)  # TODO: define the relation between precision and shots
+        estimator_v1 = Estimator(options={"seed": 1, "shots": shots})
         estimator = EstimatorV2Converter(estimator_v1)
         psi1 = self.psi[0]
         hamiltonian1 = self.hamiltonian[0]
         theta1 = self.theta[0]
         job = estimator.run([(psi1, hamiltonian1, [theta1])])
         result = job.result()
-        np.testing.assert_allclose(result[0].data.evs, [1.901141473854881])
+        print(result[0].data.evs[0])
+        np.testing.assert_allclose(result[0].data.evs, [1.5856819871431034])
         # The result of the second run is the same
         job = estimator.run([(psi1, hamiltonian1, [theta1]), (psi1, hamiltonian1, [theta1])])
         result = job.result()
-        np.testing.assert_allclose(result[0].data.evs, [1.901141473854881])
-        np.testing.assert_allclose(result[1].data.evs, [1.901141473854881])
-        # precision=0 impliese the exact expectation value
+        np.testing.assert_allclose(result[0].data.evs, [1.5856819871431034])
+        np.testing.assert_allclose(result[1].data.evs, [1.5856819871431034])
+        # precision=0 is equivalent to shots=None
+        estimator_v1 = Estimator(options={"seed": 1, "shots": None})
+        estimator = EstimatorV2Converter(estimator_v1)
         job = estimator.run([(psi1, hamiltonian1, [theta1])], precision=0)
         result = job.result()
         np.testing.assert_allclose(result[0].data.evs, [1.5555572817900956])