The heights of men in the United States are normally distributed with a mean of 69.1 inches and a standard deviation of 2.9 inches. What proportion of men are taller than 6 feet (72 inches)?
You know you need to score at least 82 points on the final exam to pass your class. After the final, you find out that the average score on the exam was 78 with a standard deviation of 7. How likely is it that you pass the class?
The Correct Answer and Explanation is :
Proportion of men taller than 6 feet
- Given data:
- Mean ((\mu)) = 69.1 inches
- Standard deviation ((\sigma)) = 2.9 inches
- Target height = 72 inches (6 feet)
- Calculate the z-score:
[
z = \frac{X – \mu}{\sigma} = \frac{72 – 69.1}{2.9} = \frac{2.9}{2.9} = 1
] - Find the proportion above z = 1:
Using a standard normal distribution table or a calculator, the proportion below (z = 1) is approximately (0.8413).
Therefore, the proportion taller than 6 feet is:
[
1 – 0.8413 = 0.1587
]
Answer: Approximately 15.87% of men are taller than 6 feet.
Likelihood of passing the class
- Given data:
- Mean ((\mu)) = 78
- Standard deviation ((\sigma)) = 7
- Target score = 82
- Calculate the z-score:
[
z = \frac{X – \mu}{\sigma} = \frac{82 – 78}{7} = \frac{4}{7} \approx 0.5714
] - Find the proportion above z = 0.5714:
Using a standard normal distribution table or a calculator, the proportion below (z = 0.5714) is approximately (0.7164).
Therefore, the proportion above (z = 0.5714) is:
[
1 – 0.7164 = 0.2836
]
Answer: There is approximately a 28.36% chance of passing the class.
Explanation:
In both problems, the z-score standardizes the values into a common scale to find probabilities using the normal distribution. The z-score tells us how many standard deviations a value is from the mean. By referencing the standard normal distribution, we determine the proportion of the population that lies above or below a specific z-score. These probabilities give a clear picture of how likely an event is to occur.