Is an unexamined life worth living?

Today's assignment was to read Bertrand Russell's Problems of Philosophy, 'Chapter XV: The value of philosophy' and the section from Plato's Apology where Socrates gives a speech defending himself against the charges of "corrupting the young, and by not believing in the gods in whom the city believes, but in other daimonia that are novel".

We were then asked to post on the question "Is an unexamined life worth living?".

My response was as follows:

To answer this question I believe it is helpful to explain what we mean by "worth". For a life to be worth living it has to either have a purpose and go some way to achieve that purpose, or have some value to someone / something.

On both these points I don't see why in principle an unexamined life could not have either purpose and/or value.

Before addressing the question of whether life has to be examined before it has worth, I would like to consider whether life has any purpose in itself.

Survival machines
Richard Dawkins argues that the universe has no purpose, at least not one that we know about. In his book 'The Selfish Gene' he argues that the fundamental purpose of life is really the survival of our genes. Our bodies are "survival machines" for our genes. He writes "survival machines began as passive receptacles for the genes, providing little more than walls to protect them from the chemical warfare of their rivals and the ravages of accidental molecular bombardment". In this sense, the purpose of life is to propagate our genes, an as such an unexamined life has as much worth as an examined life. What matters is survival.

Purpose and value is a human construct
As an atheist, I do not believe that life has any purpose other than the purpose we provide for it ourselves as rational, sentient, reflective and emotional creatures.

Dawkins goes on to explain how humans have evolved into social beings. Our brains have grown to give us the ability to think and reflect. We have gained the power to overcome our mechanistic history and organise ourselves into societies that ascribe purpose, worth and value to different things.
Socrates comes across as intellectually arrogant in The Apology. He argues that "daily to discourse about virtue, and of those other things about which you hear me examining myself and others, is the greatest good of man, and that the unexamined life is not worth living". Yet he gives us little support for this proposition, other than his eloquence.

As an example of an unexamined life that I believe is worth living I would argue that great musicians or great sportsmen who do not reflect on life, but do provide great pleasure for others, are leading lives that are worth living.