Interplanetary coronal mass ejections (ICMEs) could be classified into magnetic clouds (MCs) and non-MCs according to their magnetic field signatures, and into prominence-inside ICMEs (PIs) and non-PIs based on whether they contain colder and higher helium abundance plasmas than the solar wind. It is known that the MCs often lead to magnetic storms. However, whether or not the PIs have significant geoeffectiveness is unclear. This statistical work studies the southward interplanetary magnetic field (IMF) magnitude of the PIs, and the related magnetic storms' level. The data include the IMF and plasma moments measured by ACE and WIND, and the Dst index from 1998 to 2011. The hypothesis test based on the proportions of two groups is used to analyze 95 ICMEs related to single storms (SSs). The results show that the magnetic storms caused by the PIs mostly distribute at a strong level, while that caused by the non-PIs and by all the 95 ICMEs mostly distribute at a moderate level. The PIs have a significantly higher probability of generating SSs than the non-PIs. Moreover, the MCs containing carbon-cold and helium-enhanced materials (MC&PIs) have the highest fraction of minimum Bz, less than -11 nT. Since the MC&PIs have large-scale magnetic flux rope and prominence material, the stronger southward IMF is probably provided by the prominence. It is in accordance with the observed injection of enhanced twisted flux ropes to prominence. Therefore, the detailed eruption and propagation processes of the three-part coronal mass ejections deserve more concern from a space weather perspective.