To solve these probability questions for the normal distribution $N(10, 2)$, we need to use the concepts of normalcdf and invNorm as described. Let’s break this down:

Given Information:

• The normal distribution is $N(10, 2)$, where:
  • $\mu = 10$ (mean)
  • $\sigma = 2$ (standard deviation)

We are tasked with finding the value of $a$ that corresponds to:

1. $P(X < a) = 0.1$
2. $P(X > a) = 0.1$

Steps for Solution:

1. $P(X < a) = 0.1$

This means we are looking for the value of $a$ such that the area under the normal curve to the left of $a$ is 0.1. We can use the invNorm function to find this.

• For the normal distribution $N(10, 2)$, we input: $a = \text{invNorm}(0.1, 10, 2)$

The invNorm function gives us the $a$-value corresponding to a cumulative probability (area) of 0.1.

2. $P(X > a) = 0.1$

This means we are looking for the value of $a$ such that the area under the normal curve to the right of $a$ is 0.1. To find this, we first convert it to a cumulative probability (area to the left of $a$).

• Since $P(X > a) = 0.1$, the cumulative probability to the left of $a$ is $1 - 0.1 = 0.9$.
• So, we use: $a = \text{invNorm}(0.9, 10, 2)$

Solutions:

1. $P(X < a) = 0.1$: Solve $a = \text{invNorm}(0.1, 10, 2)$.
2. $P(X > a) = 0.1$: Solve $a = \text{invNorm}(0.9, 10, 2)$.

Do you have a calculator, or would you like me to compute these values for you?

Relative Questions:

1. What is the relationship between the z-score and the normal distribution parameters?
2. How would you calculate probabilities for a range $P(a < X < b)$ in this distribution?
3. Can you explain the difference between cumulative probability and probability density?
4. What happens to the shape of the normal distribution as standard deviation increases?
5. How would you find the z-score for a particular data value?

Tip:

Always make sure your calculator's mode is set to the correct distribution (normal, binomial, etc.) when performing statistical calculations!