For the standard normal distribution, what $z$-value corresponds to a lower-tail probability of $1\%$?

• A. $-1.96$
• B. $1.96$
• C. $-2.33$
• D. $2.33$

Answer: (C)

Verified step by step guidance

1. Understand that the problem asks for the z-value (z-score) corresponding to a cumulative probability of 1% in the lower tail of the standard normal distribution. This means we want to find the z such that \(P(Z \leq z) = 0.01\) where \(Z\) is a standard normal variable.
2. Recall that the standard normal distribution is symmetric about zero, with mean 0 and standard deviation 1. The cumulative distribution function (CDF) gives the probability that \(Z\) is less than or equal to a certain value.
3. Use a standard normal distribution table or a statistical software/calculator that provides the inverse cumulative distribution function (also called the quantile function) for the standard normal distribution. This function is often denoted as \(z = \Phi^{-1}(p)\), where \(p\) is the cumulative probability.
4. Input the cumulative probability \(p = 0.01\) into the inverse CDF function to find the corresponding z-value. This will give the z-score where the area to the left under the standard normal curve is 1%.
5. Interpret the result: since 1% is in the lower tail, the z-value will be negative, indicating it lies to the left of the mean. This z-value is the answer to the problem.