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

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

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 = (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 = (X - \mu)/(\sigma)`

`1.65 = (1.5 - 0)/(\sigma)`

`1.65\sigma = 1.5`

`\sigma = (1.5)/(1.65)`

`\sigma = 0.91`

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