For the standard normal distribution, what is the area under the curve between $z=0.0$ and $z=1.79$?

• A. $0.2266$
• B. $0.9633$
• C. $0.4066$
• D. $0.4633$

Answer: (D)

Verified step by step guidance

1. Recall that the standard normal distribution is a normal distribution with mean \(\mu = 0\) and standard deviation \(\sigma = 1\). The area under the curve between two \(z\)-values represents the probability that a standard normal random variable falls between those values.
2. Identify the two \(z\)-values given: \(z = 0.0\) and \(z = 1.79\). We want to find the area under the standard normal curve between these two points.
3. Use the standard normal cumulative distribution function (CDF), denoted as \(\Phi(z)\), which gives the area under the curve to the left of a given \(z\)-value. Specifically, \(\Phi(z)\) is the probability that the variable is less than or equal to \(z\).
4. Calculate the area between \(z=0.0\) and \(z=1.79\) by subtracting the CDF value at \(z=0.0\) from the CDF value at \(z=1.79\): \(\text{Area} = \Phi(1.79) - \Phi(0.0)\)
5. Look up the values of \(\Phi(1.79)\) and \(\Phi(0.0)\) in the standard normal table or use a calculator/software that provides the standard normal CDF. Since \(\Phi(0.0) = 0.5\), the area is \(\Phi(1.79) - 0.5\). This difference gives the probability between the two \(z\)-scores.