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When do you use normalpdf, normalcdf, and invnorm?
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Step 1:
I'll explain when to use each of these statistical functions in probability and statistics:

Step 2:
: Understanding normalpdf

- Formula: $$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}
- Used to calculate the probability density (height) at a specific point in a normal distribution - Gives the relative likelihood of a specific value occurring - Useful for visualizing the shape of a normal distribution - Typically used when you want to know the exact "height" of the probability curve at a specific point

Step 3:
: Understanding normalcdf

- Formula: $$P(a \leq X \leq b) = \int_{a}^{b} \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}} dx
- Used to calculate the probability of a value falling between two points in a normal distribution - Finds the area under the normal curve between specified limits - Useful for finding cumulative probabilities - Example: Probability of a test score between 70 and 85

Step 4:
: Understanding invnorm

- Used to find the z-score or x-value corresponding to a specific cumulative probability - Essentially the "reverse" of normalcdf - Useful for finding critical values or percentiles - Example: Finding the z-score that corresponds to the 95th percentile

Final Answer

- normalpdf: Point probability density - normalcdf: Area/probability between two points - invnorm: Finding specific values given a probability