For the standard normal distribution, what proportion of the distribution lies between $z=-1.75$ and $z=1.75$?

• A. $0.99$
• B. $0.68$
• C. $0.96$
• D. $0.50$

Answer: (C)

Verified step by step guidance

1. Understand that the problem asks for the proportion of the standard normal distribution between z = -1.75 and z = 1.75. This means we want the probability P(-1.75 < Z < 1.75) where Z follows a standard normal distribution.
2. Recall that the standard normal distribution is symmetric about zero, so the area between -1.75 and 1.75 is twice the area between 0 and 1.75.
3. Use the standard normal cumulative distribution function (CDF), denoted as \(\Phi(z)\), which gives the probability that Z is less than or equal to z. The proportion between -1.75 and 1.75 can be found by calculating \(\Phi(1.75) - \Phi(-1.75)\).
4. Since the distribution is symmetric, \(\Phi(-1.75) = 1 - \Phi(1.75)\). Therefore, the proportion can also be calculated as \(2 \times \Phi(1.75) - 1\).
5. Look up the value of \(\Phi(1.75)\) in the standard normal table or use a calculator with the normal CDF function, then apply the formula from step 4 to find the proportion between -1.75 and 1.75.