Find the​ z-score that has 66.3​% of the​ distribution's area to its right.


To find the z-score that has 66.3% of the distribution's area to its right, follow these steps:

1. The percentage given to the right of the z-score is 66.3%. Convert this to the corresponding area to the left of the z-score by subtracting from 1 (since the total area under the normal distribution curve is 1):
   \[
   1 - 0.663 = 0.337
   \]
   So, 33.7% of the distribution's area is to the left of the z-score.

2. Now, look up the z-score that corresponds to a cumulative probability of 0.337 in the standard normal distribution table or use a calculator with an inverse cumulative distribution function (like `invNorm` on a calculator).

3. Using the inverse of the standard normal cumulative distribution function, we find that:
   \[
   z \approx -0.43
   \]

Thus, the z-score that has 66.3% of the distribution's area to its right is approximately **-0.43**.

Would you like more details on how to find this using a table or calculator? Here are some related questions to deepen your understanding:

1. How do you interpret z-scores in a normal distribution?
2. What is the difference between finding the area to the right and to the left of a z-score?
3. How can you calculate the z-score for a given percentile using a calculator?
4. Why is the z-score negative when more of the area is to the right of it?
5. How does changing the mean or standard deviation affect z-scores?

**Tip:** Always check whether you need the area to the left or right of the z-score before using the table or calculator!

Find the z-score that has 66.3% of the distribution's area to its right.

Learn how to calculate the z-score that corresponds to 66.3% of the normal distribution's area to its right. This guide explains key statistical concepts, including the inverse cumulative distribution function and z-scores, and provides a step-by-step solution. Suitable for students in Grades 10-12.

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