https://file.aiquickdraw.com/mathbot/file/1729918290215gpptzcaz.jpg

Start with the normal distribution given to you in Part 2. Create and answer your own probability question using your Normal Distribution CDF. This question must be of the form where you solve for a particular value, given the probability. You should not use any of the points from (Q 16), and you may not use the mean value of your distribution. Use R to solve your problem and include all code required to obtain the solution. Do NOT use a table or your calculator.

To solve this problem, let's first establish the details we have from the cumulative distribution function (CDF) plot for the normal distribution. We can see that this CDF plot represents a normal distribution with specific points marked for reference, but without using those exact points, we’ll generate a new probability question.

Assuming the CDF represents a distribution with mean \( \mu \) and standard deviation \( \sigma \), let's use the following steps:

### 1. Define the Question

Let's say: "Given a normal distribution, what is the x-value such that the probability \( P(X \leq x) = 0.85 \)?"

In this case:
- We want to solve for \( x \) such that \( P(X \leq x) = 0.85 \).
- We do not know the mean or standard deviation directly from this image, but we can estimate it based on the CDF midpoint and use R to solve the equation.

### 2. R Code for Solving the Problem

In R, we can use the `qnorm` function to find the x-value for a specified probability, given the mean and standard deviation. However, since we're not using any midpoint values from the image (such as the mean of 570 from previous points), let’s assume an approximate mean and standard deviation. For this exercise, we’ll estimate:
- Mean (\( \mu \)) = 570
- Standard Deviation (\( \sigma \)) = 15

Here’s the R code to solve this:

```r
# Given parameters
mean_value <- 570       # assumed mean based on distribution center
std_dev <- 15           # assumed standard deviation

# Desired probability
p <- 0.85

# Calculate the x-value for P(X <= x) = 0.85
x_value <- qnorm(p, mean = mean_value, sd = std_dev)
x_value
```

### 3. Explanation of the R Code

- **`qnorm(p, mean, sd)`**: This function calculates the quantile (x-value) for the given cumulative probability \( p \), using the specified mean and standard deviation.
- **Inputs**:
  - `p = 0.85`: the cumulative probability.
  - `mean = 570` and `sd = 15`: our estimated values based on the CDF distribution.

### 4. Expected Output

The code will output an x-value corresponding to a cumulative probability of 0.85 for this normal distribution.

Would you like to see a detailed explanation of how this estimation was set up or any more details on how the normal distribution functions in R? 

### Additional Questions for Practice:

1. How can you find the z-score for a given probability without using the mean value?
2. How would you modify the problem if the cumulative probability was given as 0.95 instead of 0.85?
3. What is the relationship between the CDF midpoint and the mean of the normal distribution?
4. How can we adjust the R code if the standard deviation was unknown?
5. How do we interpret the result in terms of real-world applications of normal distribution?

### Tip

When working with the normal distribution, understanding the properties of the z-score and standard deviations can help you quickly estimate probabilities for common ranges, like 68%, 95%, and 99.7%.

Learn how to solve a probability question using the normal distribution's cumulative distribution function (CDF). This guide provides R code to find an x-value corresponding to a given probability without using a table or calculator. Perfect for students in Grades 10-12.

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