Question

A signal in a communication channel is called a false signal if its voltage is higher than 1.5 volts in absolute value. (Note that the voltage can be positive or negative here.) Assume that the voltage of a signal is normally distributed with a mean of 0. What is the standard deviation of voltage such that the probability of a false signal is 0.005.

Answer 1

Answer:

The standard deviation is 0.5338V.

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:

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

The Z-score measures how many standard deviations the measure is from the mean. 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, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 0, \sigma = ?$

What is the standard deviation of voltage such that the probability of a false signal is 0.005.

This means that when X = 1.5, Z has a pvalue of 1-(0.005/2) = 0.9975. So when X = 1.5, Z = 2.81. Also when X = -1.5, Z has a pvalue of 0.0025. So Z = -2.81. Applying any of then, we find the standard deviation.

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

$2.81 = \frac{1.5 - 0}{\sigma}$

$2.81\sigma = 1.5$

$\sigma = \frac{1.5}{2.81}$

$\sigma = 0.5338$

The standard deviation is 0.5338V.

Answer 2

Answer:

σ =0.5814

Step-by-step explanation:

Let x denote the voltage and x follow normal random variable with μ = 0 and σ = ?

Given that x > 1.5, signal communication i detected.

the probability  of false signal is 0.005.

false signal only occurs when x < 1.5

therefore: P(x ≤ 1.5) = 0.005

→ (1.5 -0)/σ = 1/∅*0.005 = 2.58

→ 2.58σ = 1.5

therefore, σ = 0.5814