Heights - Normal Distribution
Assume that adult heights are normally distributed with a mean of 175cm and a standard deviation of 7cm. What is the probability that a randomly selected adult is taller than 182cm?
Related Concepts
Hint
To solve this, we'll follow these steps:
- Identify the given information:
- The heights follow a Normal distribution.
- Mean (μ) = 175cm.
- Standard deviation (σ) = 7cm.
- We are interested in the height X = 182cm.
- Calculate the Z-score: Convert the height 182cm into a Z-score. The Z-score tells us how many standard deviations 182cm is from the mean of 175cm. Use the formula:
Z = (X - μ) / σ - Determine what probability you need: We want the probability that an adult is taller than 182cm. In terms of Z-scores, this means P(Z > your calculated Z-score).
- Use a Z-table (or calculator):
- Standard Z-tables usually give the area to the left of a Z-score (i.e., P(Z ≤ z)). Let's call this value Φ(z).
- Since we want the area to the right (P(Z > z)), we use the fact that the total area under the curve is 1. So, P(Z > z) = 1 - P(Z ≤ z) = 1 - Φ(z).
Think about it: 182cm is above the average of 175cm. How many standard deviations above?
Explanation: Heights - Normal Distribution
Imagine we've measured the heights of many adults. If we plot these heights on a graph, they often form a "bell curve" shape. This is called a Normal distribution.
- The peak of the bell is at the average height, which is 175cm in this problem.
- The spread of the bell (how wide or narrow it is) is described by the standard deviation, which is 7cm here. A smaller standard deviation means most people are very close to the average height; a larger one means heights are more varied.
Our Question: What's the chance a randomly picked adult is taller than 182cm?
Step-by-Step for Beginners:
- How far is 182cm from the average? The average (μ) is 175cm. The height we're interested in (X) is 182cm. The difference is 182cm - 175cm = 7cm. So, 182cm is 7cm above the average.
- How many "standard steps" (standard deviations) is that? One "standard step&" (σ) is 7cm. Since 182cm is 7cm above the average, it's exactly 7cm / 7cm = 1 standard step above the average. This "number of standard steps" is called the Z-score. So, for a height of 182cm, the Z-score is 1.
- What does a Z-score of 1 mean? A Z-score of 1 means this height is higher than a good portion of people, but not extremely rare.
- Finding the probability (the chance): We use something called a Z-table (or a calculator with this function). This table tells us the percentage of people whose Z-score is less than or equal to the one we calculated. For a Z-score of 1, the table tells us that about 0.8413 (or 84.13%) of people have a Z-score of 1 or less (meaning they are 182cm or shorter).
- Getting our answer: We want the chance of being *taller than* 182cm. If 84.13% are 182cm or shorter, then the rest must be taller. The total chance is 1 (or 100%). So, the chance of being taller is 1 - 0.8413 = 0.1587. This means there's about a 15.87% chance that a randomly selected adult is taller than 182cm.
Let X be the random variable representing adult heights. We are told that X follows a Normal distribution with:
- Mean (μ) = 175 cm
- Standard Deviation (σ) = 7 cm
We can write this as X ~ N(175, 7²).
We want to find the probability that a randomly selected adult is taller than 182cm, i.e., P(X > 182).
1. Standardize the Value (Calculate the Z-score)
To use standard normal tables or functions, we first convert the height X = 182cm into a Z-score. The Z-score tells us how many standard deviations a particular value is from the mean.
The formula for the Z-score is: Z = (X - μ) / σ
Plugging in our values: Z = (182 - 175) / 7 Z = 7 / 7 Z = 1
This Z-score of 1 means that a height of 182cm is exactly 1 standard deviation above the average height.
2. Find the Probability using the Z-score
We are looking for P(X > 182). After standardizing, this is equivalent to finding P(Z > 1), where Z is a standard normal variable (mean 0, standard deviation 1).
Standard Z-tables (or statistical calculators) usually give the cumulative probability P(Z ≤ z), which is the area under the bell curve to the left of the Z-score 'z'. This is often denoted as Φ(z).
To find the area to the right, P(Z > 1), we use the property that the total area under the curve is 1: P(Z > 1) = 1 - P(Z ≤ 1) P(Z > 1) = 1 - Φ(1)
Using a Z-table or a calculator for Φ(1) (the probability that a standard normal variable is less than or equal to 1): Φ(1) ≈ 0.84134
Now, we can calculate P(Z > 1): P(Z > 1) ≈ 1 - 0.84134 P(Z > 1) ≈ 0.15866
Final Result
The probability that a randomly selected adult is taller than 182cm is approximately 0.1587 (when rounded to four decimal places).
P(Height > 182cm) ≈ 0.1587
This means there is about a 15.87% chance that a randomly selected adult from this population will be taller than 182cm.
Visualizing the Bell Curve: Imagine the bell curve centered at 175cm. The value 182cm (which is Z=1) is to the right of the center. The Z-table gives you the area from the far left tail up to Z=1 (which is ~0.8413). We are interested in the area in the right tail, beyond Z=1. So, we subtract the area to the left from the total area (1) to get the area to the right.
The shaded area represents P(Z > 1).
Explore Further: Using the same distribution (mean=175cm, std. dev=7cm), what is the probability that a randomly selected adult's height is between 168cm and 182cm?
(Hint: This is P(168 < X < 182). You'll need to find Z-scores for both 168cm and 182cm. Then, the probability is P(Z168 < Z < Z182) = Φ(Z182) - Φ(Z168)).