Renna's statistics project involves analyzing the heights of American females, which are normally distributed with a mean of 64 inches and a standard deviation of 3 inches. With 1,000 women enrolled at her community college, Renna wants to know how many of them are expected to be 67 inches and 70 inches tall. This arti...
Normal Distribution
The normal distribution is a bell-shaped curve that represents a continuous probability distribution. In this case, it describes the distribution of heights among American females. The mean (64 inches) represents the center of the distribution, and the standard deviation (3 inches) indicates how spread out the data is.
Z-Scores
To use the normal distribution graph to determine probabilities, we need to convert the heights to z-scores. A z-score represents the number of standard deviations a data point is away from the mean. It is calculated using the following formula:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the data point (height in this case)
- μ is the mean (64 inches)
- σ is the standard deviation (3 inches)
Calculating Z-Scores for Renna's Problem
For a height of 67 inches:
z = (67 - 64) / 3 = 1
For a height of 70 inches:
z = (70 - 64) / 3 = 2
Using the Normal Distribution Graph
The normal distribution graph shows the probabilities associated with different z-scores. The area under the curve between two z-scores represents the probability of a data point falling within that range.
To determine the probability of a woman being between 67 and 70 inches tall, we need to find the area under the curve between z-scores of 1 and 2. This area represents the proportion of women within that height range.
Finding the Area Under the Curve
You can use a standard normal distribution table or statistical software to find the area under the curve. The table gives the cumulative probability for each z-score, meaning the area under the curve to the left of that z-score.
To find the area between z-scores of 1 and 2, subtract the cumulative probability for z = 1 from the cumulative probability for z = 2.
Calculating Expected Number of Women
Once you have the probability of a woman being between 67 and 70 inches tall, multiply that probability by the total number of women enrolled at the community college (1,000) to get the expected number of women within that height range.
Example Calculation
Let's say the area under the curve between z-scores of 1 and 2 is 0.1359. This means 13.59% of American females are between 67 and 70 inches tall.
To find the expected number of women at the community college within that height range, multiply the probability by the total number of women:
Expected number of women = 0.1359 * 1000 = 135.9
Therefore, Renna would expect approximately 136 women at her community college to be between 67 and 70 inches tall.
Conclusion
By understanding the normal distribution and using z-scores, Renna can determine the expected number of women at her community college within specific height ranges. This process involves converting heights to z-scores, finding the area under the normal distribution curve between those z-scores, and multiplying that probability by the total number of women enrolled. By following these steps, Renna can successfully complete her statistics project.