Find probabilities for the standard and non-standard normal distribution.

You may round your answers to four decimal places

a.

0.6026

b.

c.

d. If
= 68 and = 12, find 

Let's break this down step by step to compute the probabilities for both standard and non-standard normal distributions. You mentioned probabilities for the values a-d but didn't specify the complete context or provide full values for some parts of the question. I'll do my best with the information you've shared:

### Part a: Probability for \( Z = 0.6026 \) (Standard Normal Distribution)
For the standard normal distribution, where \( Z \) follows a standard normal distribution (mean \( \mu = 0 \) and standard deviation \( \sigma = 1 \)):

1. **Look up the Z-score of 0.6026** in the standard normal table or use a calculator.

The corresponding probability is:
\[
P(Z \leq 0.6026) \approx 0.7264
\]
So, \( P(Z \leq 0.6026) \approx 0.7264 \), meaning there's a 72.64% probability that a value drawn from the standard normal distribution will be less than or equal to 0.6026.

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### Part b: You left this part blank
Could you provide the value or clarification for part b? I'd be happy to compute it once I have the needed details.

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### Part c: You left this part blank as well
Please let me know what you'd like to solve here. We could compute another Z-score probability or something related to the non-standard normal distribution.

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### Part d: Non-Standard Normal Distribution
You mentioned:

If \( \mu = 68 \) and \( \sigma = 12 \), find...
It seems like you want to find a Z-score or probability based on these values. To calculate the Z-score or probability for a non-standard normal distribution, we use the formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

Could you specify the value of \( X \), or are you looking for the probability of being above or below a certain value?

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Feel free to fill in the gaps for parts b, c, and d, and I'll continue the calculations!

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Do you have any questions or would you like further explanation on the process?

### Related Questions:
1. How do you calculate probabilities for negative Z-scores?
2. What is the relationship between the cumulative distribution function and Z-scores?
3. How do you standardize a normal variable to compute probabilities?
4. What is the empirical rule (68-95-99.7 rule) in relation to normal distributions?
5. How do we find probabilities for non-standard normal distributions without tables?

#### Tip:
Always standardize your normal variable (i.e., compute the Z-score) before using the standard normal table, as it assumes \( \mu = 0 \) and \( \sigma = 1 \).

Find probabilities for the standard and non-standard normal distribution. You may round your answers to four decimal places.
a.
0.6026
b.
c.
d. If µ = 68 and σ = 12, find ...

Learn how to calculate probabilities for both standard and non-standard normal distributions. This solution covers Z-scores, the empirical rule, and includes step-by-step examples with a given value of 0.6026. Suitable for high school students in Grades 10-12.

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