(b) The probability a male's growth plates fuse before age 17 is approximately 0.0584.
(c) The proportion of male growth plates fusing between 16 and 17 years of age is approximately 0.0827.
(d) Yes, it would be considered unusual for a male's growth plates to fuse when he is 23 years old or older due to the low probability.
How to solve probabilities
To solve these questions, use the normal distribution with the given mean and standard deviation. Let's calculate each part step by step:
To find the probability a male's growth plates fuse before age 17, calculate the z-score and then find the corresponding cumulative probability.
First, convert the given age of 17 years into months, which is 17 * 12 = 204 months.
Next, calculate the z-score using the formula:
z = (x - μ) / σ
where x is the value we are interested in (204 months), μ is the mean (19.1 years * 12 = 229.2 months), and σ is the standard deviation (16.1 months).
z = (204 - 229.2) / 16.1
Calculating this, we find:
z ≈ -1.5665
Now, find the cumulative probability using the z-score:
P(z < -1.5665) ≈ 0.0584
Therefore, the probability a male's growth plates fuse before age 17 is approximately 0.0584.
The probability a male's growth plates fuse between 16 and 17 years of age can be found by calculating the cumulative probability between the corresponding z-scores.
First, convert the ages 16 and 17 years into months:
16 years * 12 = 192 months
17 years * 12 = 204 months
Next, calculate the z-scores for both ages:
z₁ = (192 - 229.2) / 16.1
z₂ = (204 - 229.2) / 16.1
Calculating these, we find:
z₁ ≈ -2.3068
z₂ ≈ -1.5665
Now, find the difference between the cumulative probabilities:
P(-2.3068 < z < -1.5665)
Using a standard normal distribution table or a calculator, we find:
P(-2.3068 < z < -1.5665) ≈ 0.0827
Therefore, the proportion of male growth plates fusing between 16 and 17 years of age is approximately 0.0827.
(d) To determine if it would be unusual for a male's growth plates to fuse when he is 23 years old or older, calculate the z-score for the age of 23 years and find the corresponding cumulative probability.
Converting 23 years into months:
23 years * 12 = 276 months
Calculating the z-score:
z = (276 - 229.2) / 16.1
Calculating this, we find:
z ≈ 2.9068
Now, find the cumulative probability:
P(z > 2.9068) ≈ 0.0018
Since the resulting probability is very small (less than 0.05), it would be considered unusual for a male's growth plates to fuse when he is 23 years old or older.