Termo relay on arduino i need help

[EmilyJane]

@jim-p

 // Read the thermistor voltage num_samples times
  for (i = 0; i < num_samples; i++)
  {
    //samples += analogRead(Vout_pin);
    // for testing
    samples += 974;

    delay(10);
  }

Since @Jim-p is done for the day, I'll give you a hint.
The above code that he posted was just a test. If you change it to this:

 // Read the thermistor voltage num_samples times
  for (i = 0; i < num_samples; i++)
  {
    samples += analogRead(Vout_pin);
    // for testing
   // samples += 974;

    delay(10);
  }

You'll see what your circuit does.