The problem asks to find the probability that a given score is less than 1.05 using a normal distribution. Here's a breakdown of the situation based on the image:

1. The scores are normally distributed with a mean of 0 and a standard deviation of 1.
2. The task is to determine which graph correctly represents the cumulative area under the curve to the left of $x = 1.05$, which corresponds to finding $P(X < 1.05)$.

Standard Normal Distribution

For a standard normal distribution (mean = 0, standard deviation = 1), we need to find the cumulative probability for $z = 1.05$.

Using the Z-score table, $P(Z < 1.05) \approx 0.8531$.

This means that about 85.31% of the data lies below a Z-score of 1.05, so the correct graph should show an area under the curve to the left of $x = 1.05$ that covers about 85% of the total area.

Graph Interpretation

Based on the question:

• The correct graph should shade a large portion of the left side of the curve (about 85%).
• The peak should be centered at $x = 0$, with a standard bell curve extending on both sides.

Graph C appears to match this description since it shows most of the area under the curve shaded to the left of $x = 1.05$, which is consistent with the solution from the Z-score table.

Answer:

The correct option is C.

Would you like further details on how this problem is solved or explanations of related concepts?

Here are five related questions you might find useful:

1. How do you use a Z-score table to find probabilities?
2. What is the standard normal distribution, and why is it important?
3. How do you calculate the Z-score for a data point from any normal distribution?
4. How do cumulative probabilities work in a normal distribution?
5. What is the significance of the area under the curve in probability?

Tip: When working with normal distributions, the Z-score allows you to compare values from any normal distribution to the standard normal distribution.