Let's go through the questions one by one based on the image provided:

Problem 6: Find the probability that $x$ falls in the shaded area.

The image shows a uniform distribution from $x = 3$ to $x = 5$, with a probability density of $\frac{1}{10}$.

To find the probability, use the formula for uniform distribution probability: $P(a \leq x \leq b) = (b - a) \times \text{density}$ In this case:

• The interval $a = 3$ and $b = 5$,
• The density is $\frac{1}{10}$.

So, the probability is: $P(3 \leq x \leq 5) = (5 - 3) \times \frac{1}{10} = 2 \times \frac{1}{10} = 0.2$ Thus, the probability that $x$ falls in the shaded area is 0.2 or 20%.

Problem 7: What percent of students scored between 60 and 90 points?

The distribution is approximately Normal with:

• Mean $\mu = 75$,
• Standard deviation $\sigma = 7.5$.

We calculate the z-scores for 60 and 90 points using the formula: $z = \frac{x - \mu}{\sigma}$

• For 60: $z = \frac{60 - 75}{7.5} = \frac{-15}{7.5} = -2$
• For 90: $z = \frac{90 - 75}{7.5} = \frac{15}{7.5} = 2$

Using the standard normal distribution table:

• The area to the left of $z = -2$ is approximately 0.0228,
• The area to the left of $z = 2$ is approximately 0.9772.

Thus, the percentage of students scoring between 60 and 90 is: $0.9772 - 0.0228 = 0.9544 \quad \text{or} \quad 95.44\%$ So, approximately 95.44% of the students scored between 60 and 90 points.

Problem 8: What percent of students failed the test (scored below 60)?

Using the $z$-score for 60 (calculated earlier as $z = -2$):

The area to the left of $z = -2$ is approximately 0.0228, meaning about 2.28% of the students scored below 60 points.

Problem 9: What percent of students scored below 70?

For 70 points: $z = \frac{70 - 75}{7.5} = \frac{-5}{7.5} = -0.67$ Using the standard normal table, the area to the left of $z = -0.67$ is approximately 0.2514.

Thus, about 25.14% of the students scored below 70 points.

Do you want more details or have any questions about these steps? Here are some related questions to deepen understanding:

1. How would the probability change in Problem 6 if the shaded area was from $x = 2$ to $x = 4$?
2. What is the probability that a student scored above 90 points?
3. If the standard deviation increased to 10, how would that affect the percentage of students scoring between 60 and 90 points?
4. How can you use the empirical rule to approximate the answer for students scoring between 60 and 90?
5. What is the probability that a student scored between 70 and 85 points?

Tip: Using the z-score is essential for calculating probabilities in a normal distribution! Make sure to get comfortable with it.