What is the formula of the Kth percentile of X when the z-score is known?Question 1Answera.k=μ+σb.k=μ-σc.k=μ+z*σ

To find the formula for the Kth percentile of a given variable $X$ when the z-score is known, here are the steps to break down and analyze the question.

1. ### Break Down the Problem

1. Identify what a percentile is and how it relates to a distribution.
2. Understand the role of the z-score in the context of a normal distribution.
3. Recognize the formula related to converting a z-score to a value in the original distribution.

2. ### Relevant Concepts

• The Kth percentile $P_k$ of a normal distribution can be expressed using the z-score: $P_k = \mu + z \cdot \sigma$ where:
  • $P_k$ is the Kth percentile,
  • $\mu$ is the mean of the distribution,
  • $\sigma$ is the standard deviation,
  • $z$ is the z-score corresponding to the Kth percentile.

3. ### Analysis and Detail

1. The z-score represents how many standard deviations an element is from the mean.
2. To find the Kth percentile using the z-score, you take the mean $\mu$ and add the product of the z-score and the standard deviation $\sigma$.
3. This allows one to locate the Kth percentile in a normally distributed set of data.

4. ### Verify and Summarize

• The provided option $c.k = \mu + z \cdot \sigma$ is indeed the correct choice for finding the Kth percentile using the z-score.

Final Answer

The formula for the Kth percentile $P_k$ of $X$ when the z-score is known is given by: $P_k = \mu + z \cdot \sigma$ Therefore, the correct answer is c. $k = \mu + z \cdot \sigma$.