MATH SOLVE

2 months ago

Q:
# 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?

Accepted Solution

A:

Answer:The standard deviation of the voltage is [tex]\sigma = 0.91[/tex]Step-by-step explanation:Problems of normally distributed samples can be solved using the z-score formula.In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by[tex]Z = \frac{X - \mu}{\sigma}[/tex]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 thatAssume that the voltage is normally distributed with a mean of 0, so [tex]\mu = 0[/tex]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 [tex]P(X>1.5) = 0.05[/tex]. This means that when [tex]X = 1.5[/tex] the zscore has a pvalue of 0.95. Looking at the zscore table, we have that [tex]Z = 1.65[/tex]. So[tex]Z = \frac{X - \mu}{\sigma}[/tex][tex]1.65 = \frac{1.5 - 0}{\sigma}[/tex][tex]1.65\sigma = 1.5[/tex][tex]\sigma = \frac{1.5}{1.65}[/tex][tex]\sigma = 0.91[/tex]The standard deviation of the voltage is [tex]\sigma = 0.91[/tex]