Question

A signal in a communication channel is detected when the voltage is higher than 1.5 volts in absolute value. Assume that the voltage is normally distributed with a mean of 0. What is the standard deviation of voltage such that the probability of a false signal is 0.05?

Answer 1

Answer:

The standard deviation of the voltage is $\sigma = 0.91$

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by

$Z = \frac{X - \mu}{\sigma}$

After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X. Subtracting 1 by the pvalue, we This p-value is the probability that the value of the measure is greater than X.

The problem states that

Assume that the voltage is normally distributed with a mean of 0, so $\mu = 0$

A signal in a communication channel is detected when the voltage is higher than 1.5 volts in absolute value. What is the standard deviation of voltage such that the probability of a false signal is 0.05?

We know that $P(X>1.5) = 0.05$. This means that when $X = 1.5$ the zscore has a pvalue of 0.95. Looking at the zscore table, we have that $Z = 1.65$. So

$Z = \frac{X - \mu}{\sigma}$

$1.65 = \frac{1.5 - 0}{\sigma}$

$1.65\sigma = 1.5$

$\sigma = \frac{1.5}{1.65}$

$\sigma = 0.91$

The standard deviation of the voltage is $\sigma = 0.91$