Smart speakers (such as Amazon Echo and Google Home) are becoming a popular addition to homes, and it is expected that they will be used as parental support in the future. Previous studies have analyzed the history logs of the uses of their services and gathered impressions of them using questionnaire surveys and qualitative interviews. Details of how children interact with smart speakers have not been fully investigated. This study investigated children's interactions with an utterance-output device, such as a smart speaker, and focuses on the influences of the children's personality traits. The experiment was conducted in a realistic setting. The results indicate that the less nervous, more emotionally stable, or more adaptable to communication different from that at home children are, the more closely they engage in conversational play with an utterance-output device. The findings suggest that effective conversational play, which can lessen anxiety toward a novel utterance-output device, will be required in future interaction design.