Question

Let \(z\) denote a random variable having a normal distribution with \(\mu=0\) and \(\sigma=1\). Determine each of the following probabilities: a. \(P(z<0.10)\) b. \(P(z<-0.10)\) c. \(P(0.40-1.25)\) g. \(P(z<-1.50\) or \(z>2.50)\)

Short answer

The short answer will be the specific values calculated for each of the probabilities from parts a through g using the standard normal distribution table or calculator.

Step-by-step solution

Understanding the Normal Distribution CDF

The cumulative distribution function (CDF) gives the probability that a random variable is less than or equal to a certain value. For a normally distributed random variable, it is typically represented as \(P(z < x)\), where \(z\) is the standard normal variable and \(x\) is the value of interest.

Calculating Probabilities using the CDF

Using the normal distribution tables or a calculator with a built-in normal distribution function, the probabilities can be calculated as follows: a. \(P(z<0.10) = F(0.10)\) b. \(P(z<-0.10) = F(-0.10)\) c. \(P(0.40-1.25) = 1 - F(-1.25)\) g. \(P(z<-1.50\) or \(z>2.50) = F(-1.50) + (1 - F(2.50))\)\nNote: F(x) represents the value of the cumulative distribution function at x.

Using the Cumulative Distribution Function (CDF) Values

Replace each instance of F(x) with the appropriate value from the standard normal distribution table or calculator. Follow the order of operations to carry out the arithmetic.