When working on a normal distribution other than the standard normal distribution you need to either first convert to the z-scores and then find probabilities using the standard normal distribution(N(0,1)) OR when using technology you can input the mean and standard deviation.

P (a < X < b) = normalcdf(a,b,)=normalcdf(staring data value, ending data value, mean, standard deviation) to find the probability given the interval of data values.

This is equal to P(<Z<) = normalcdf(,)=normalcdf(smaller z-score, larger z-score)

For finding the data value given the area you can use invNorm. Again you either need to convert the z-score back to the data value OR tell the calculator the mean and standard deviation.

invNorm(area to the left, mean, standard deviation) = data value =X

invNorm(area to the left) = Z = Z-score. Then X =

For the distribution N(10,2) match the probabilities below:

Group of answer choices
P(X < a) = 0.1

P(X > a) = 0.1

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!

When working on a normal distribution other than the standard normal distribution you need to either first convert to the z-scores and then find probabilities using the standard normal distribution(N(0,1)) OR when using technology you can input the mean and standard deviation.

P (a < X < b) = normalcdf(a,b,)=normalcdf(starting data value, ending data value, mean, standard deviation) to find the probability given the interval of data values.

This is equal to P(<Z<) = normalcdf(,)=normalcdf(smaller z-score, larger z-score)

For finding the data value given the area you can use invNorm. Again you either need to convert the z-score back to the data value OR tell the calculator the mean and standard deviation.

invNorm(area to the left, mean, standard deviation) = data value =X

invNorm(area to the left) = Z = Z-score. Then X =

For the distribution N(10,2) match the probabilities below:

Group of answer choices
P(X < a) = 0.1

P(X > a) = 0.1

Learn how to solve probability questions for the normal distribution N(10, 2) by using the invNorm and normalcdf functions. This guide explains how to calculate P(X < a) and P(X > a) and highlights key concepts such as Z-scores and cumulative probabilities. Suitable for Grades 10-12.

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