I'll analyze the image and provide a step-by-step guide to solve the problems. PIL import Image from pytesseract import image_to_string

Load the image to extract text

image_path = '/mnt/data/1740674074418uv4az3mj.jpg' image = Image.open(image_path)

Use OCR to extract text from the image

extracted_text = image_to_string(image)

extracted_textThe extracted text is not perfectly clear, but here's a step-by-step guide to solving the problems in the image:

Part A: Illustrating Areas Under the Standard Normal Curve

1. The area from $z = 0$ to $z = 1.23$:
   • The area under the standard normal curve from 0 to 1.23 can be found using the standard normal distribution table or a calculator:
   • $P(z \leq 1.23) = 0.8907$.
   • The area from $z = 0$ to $z = 1.23$ is $0.8907 - 0.5 = 0.3907$ (since 0 to the mean is 0.5).
   • The answer is approximately 0.39 or 39%.

2. The area from $z = -1.05$ to $z = 0.5$:
   • $P(z \leq -1.05) = 0.1469$.
   • $P(z \leq 0.5) = 0.6915$.
   • The area between $z = -1.05$ and $z = 0.5$ is:
   • $P(-1.05 < z < 0.5) = 0.6915 - 0.1469 = 0.5446$.
   • The answer is 0.5446 or 54.46%.

3. The area from $z = 2.45$ to $z = 1$:
   • $P(z \leq 2.45) = 0.9929$.
   • $P(z \leq 1) = 0.8413$.
   • The area between $z = 1$ and $z = 2.45$ is:
   • $P(1 < z < 2.45) = 0.9929 - 0.8413 = 0.1516$.
   • The answer is 0.1516 or 15.16%.

4. The area above $z = -0.3$:
   • $P(z \leq -0.3) = 0.3821$.
   • The area above $z = -0.3$ is:
   • $P(z > -0.3) = 1 - 0.3821 = 0.6179$.
   • The answer is 0.6179 or 61.79%.

5. The area to the left of $z = -1.23$:
   • $P(z \leq -1.23) = 0.1093$.
   • The answer is 0.1093 or 10.93%.

Part B: Probability of Z-Scores

1. Falls between $z = 0$ to $z = 1.25$:
   • $P(z \leq 1.25) = 0.8944$.
   • The area from $z = 0$ to $z = 1.25$ is:
   • $P(0 < z < 1.25) = 0.8944 - 0.5 = 0.3944$.
   • The answer is 0.3944 or 39.44%.

2. Falls above $z = -2.1$:
   • $P(z \leq -2.1) = 0.0179$.
   • The area above $z = -2.1$ is:
   • $P(z > -2.1) = 1 - 0.0179 = 0.9821$.
   • The answer is 0.9821 or 98.21%.

Additional Practice Questions

1. What is the probability of a z-score being below $z = 0.7$?
2. How do you find the area between two z-scores, like $z = -1.5$ and $z = 0.8$?
3. What does it mean if a z-score is 2 standard deviations above the mean?
4. How can you use a z-table to find the area in the tail of the distribution?
5. How do you interpret a z-score of 0?

Tip:

When finding areas under the standard normal curve, remember that the total area under the curve is 1, and it is symmetrical around 0.