A normal distribution has u 32 and o = 5. (a) Find the z score corresponding to x = 27. (b) Find the z score corresponding to x = 43. (c) Find the raw score corresponding to z = -2. (d) Find the raw score corresponding to z = 1.5 %3D
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![### Understanding Z-Scores and Raw Scores in a Normal Distribution
A normal distribution is specified with a mean (\(\mu\)) of 32 and a standard deviation (\(\sigma\)) of 5. Below are exercises to calculate z-scores and raw scores based on given values.
#### Exercises:
(a) **Find the z-score corresponding to \(x = 27\).**
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(b) **Find the z-score corresponding to \(x = 43\).**
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(c) **Find the raw score corresponding to \(z = -2\).**
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(d) **Find the raw score corresponding to \(z = 1.5\).**
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#### Explanation:
- **Z-Score Calculation**: The z-score tells you how many standard deviations a value is from the mean. It is calculated using the formula:
\[
z = \frac{{x - \mu}}{\sigma}
\]
- **Raw Score Calculation**: The raw score is the actual value in the data set. To find it from a z-score, use:
\[
x = z \cdot \sigma + \mu
\]
These exercises are designed to help you practice and understand the relationship between a data point, the mean, and the standard deviation in a normal distribution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2003d332-f35a-4e8d-8a41-5d3fb7e691ff%2Ff29565fd-7aa3-4abf-8683-14882e4795f1%2F0g75u4_processed.jpeg&w=3840&q=75)
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