Noise Math Example 4

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Example 4

hard
In a medical trial, the treatment shows an improvement of 3 points on a pain scale, but each patient's response varies with SD = 8. With n=25n = 25 patients, calculate the signal-to-noise ratio (SNR = effect/SE) and determine if the signal can be detected.

Solution

  1. 1
    Standard error: SE=ฯƒn=825=85=1.6SE = \frac{\sigma}{\sqrt{n}} = \frac{8}{\sqrt{25}} = \frac{8}{5} = 1.6
  2. 2
    Signal-to-noise ratio: SNR=effectSE=31.6=1.875SNR = \frac{\text{effect}}{SE} = \frac{3}{1.6} = 1.875
  3. 3
    This corresponds to a zz-score of 1.875; P(Z>1.875)โ‰ˆ0.030P(Z > 1.875) \approx 0.030, so p-value โ‰ˆ 0.030
  4. 4
    At ฮฑ=0.05\alpha = 0.05: the signal (3-point improvement) is detectable above the noise with n=25

Answer

SNR โ‰ˆ 1.875; p-value โ‰ˆ 0.030; the treatment effect is statistically detectable.
Signal-to-noise ratio in statistics is the effect size divided by standard error. Increasing sample size decreases SE (noise), making it easier to detect real effects (signals). This is why power analysis matters in study design.

About Noise

Noise is random variation in data that is not explained by the underlying pattern or model โ€” the unpredictable fluctuations around the true signal.

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