Rashid is 84 inches tall. If the height of males in the United States is nor- mally distributed with a mean of 70 inches and a standard deviation of 4 inches, what is the z-score of Rashid's height?
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 in a Normal Distribution
In this example, we will calculate the z-score of Rashid's height, given that the height of males in the United States is normally distributed with a specific mean and standard deviation.
**Problem:**
Rashid is 84 inches tall. If the height of males in the United States is normally distributed with:
- a mean (\(\mu\)) of 70 inches
- a standard deviation (\(\sigma\)) of 4 inches
What is the z-score of Rashid's height?
**Solution:**
To calculate the z-score of Rashid's height, we use the following formula:
\[ z = \frac{(X - \mu)}{\sigma} \]
Where:
- \( z \) is the z-score
- \( X \) is the value we are standardizing (Rashid's height, which is 84 inches)
- \( \mu \) is the mean of the distribution (70 inches)
- \( \sigma \) is the standard deviation of the distribution (4 inches)
Plug in the values:
\[ z = \frac{(84 - 70)}{4} \]
\[ z = \frac{14}{4} \]
\[ z = 3.5 \]
**Interpretation:**
A z-score of 3.5 indicates that Rashid's height is 3.5 standard deviations above the mean height of males in the United States. This means Rashid's height is significantly higher than the average male height in the United States.
### Key Points:
- The z-score allows us to understand how far a specific value is from the mean in terms of standard deviations.
- A higher positive z-score indicates a value above the mean, while a negative z-score would indicate a value below the mean.
- Understanding z-scores helps in comparing data points from different distributions or sets of data.
Feel free to use this calculation method to determine z-scores for other sets of data that follow a normal distribution!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F81d865ca-ed2c-4f6d-a4cc-7f843242dec1%2F730a815d-23db-4527-aa69-fff50434b912%2F777c8zr_processed.jpeg&w=3840&q=75)
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