Is it better for females to pray at home during Umrah/Hajj?

Al Jawaab Billahit-Tawfeeq

Hadhrat Anas bin Malik (may Allah be pleased with him) reported that Allaah’s Messenger (may Allah’s blessings and peace be upon him) said: Prayer of a man in his house is one prayer, and his prayer in the mosque of the locality is prayer by 25 times, and his prayer in the congregational mosque is prayer by 500 times, and his prayer in the farthest mosque Masjid Aqsa (of Jerusalem) is prayer by 50,000 times and his prayer in my mosque is prayer by 50,000 times and his prayer in the sacred mosque (of Ka’ba) is prayer by 100,000 times. (Ibn Majah)

The Prophet (may Allah’s blessings and peace be upon him) said, “He who performs forty Salaah in my Masjid, in such a way that he does not miss a single Salaah, Allaah prescribes for him freedom from the fire, freedom from punishment and freedom from hypocrisy. (Musnad Ahmad)

The Ahaadeeth clearly point out the great merits of performing Salaah in the Sacred Masaajid. However as you have pointed out, it is best for a woman to perform Salaah in her hotel room. She will receive the same amount of reward as the men.

It is narrated Umm Humaid As-Sa’diyah (radhiallahu ‘anha) who once said to the Prophet (sallallahu ‘alaihi wasallam): Messenger of Allah! I love to pray with you (i.e. in the masjid-e-nabawi). Upon hearing this, Allah’s Messenger (sallallahu ‘alaihi wasallam) said:
‘I know that you love to say your prayer with me. However, the prayer you offer in your residence is more excellent than that you offer in your locality’s masjid. And your prayer in your locality’s masjid is better for you than you prayer in the Jami’ Masjid (larger congregational masjid).’ (Abu Dawud and At-Tabaraani)

NOTE: It will be abominable if a woman performs Salah in the Haram, and in the interim the rules of Hijab are breached, or she is unable to avoid colliding with males due to the big crowds.

And Allah knows best

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A. Z. Pandor