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 $\sigma = 1$.