The age of the Earth can be calculated from the Hubble constant (H), that is, the ratio between the recession speed of a galaxy, quasar, etc. and its distance. The value of the Hubble constant is about 69.8 km/s/megaparsec. The ratio of the speed of light (c) to H is about 4300 parsecs, corresponding to an age of 14.0 billion years.
Also, the Big Bang cosmology predicts both the existence and the non-uniformity of the cosmic microwave background; the observed non-uniformity (measured by the Wilkinson Microwave Anisotropy Probe - WMAP - and the ESA Planck satellite) corresponds to an age of 13.8 billion years, in good agreement with the age derived from the Hubble constant.
The time between the origin of the universe and the recombination epoch (about 380,000 years) is calculated from the observed value of the Hubble constant.
Although the Big Bang cosmology has not been proved (no scientific theory can ever be proved, since it is always possible that new observations will disprove it), it explains the observations better than any alternative hypothesis so far proposed. If you can put forward a different hypothesis and can show that it explains the facts better than the Big Bang cosmology, you will become famous overnight.