Answer :
Final answer:
In this case, the conditional probability P(X>120|X>105) of the serum zinc level being greater than 120 given that it is already known to be greater than 105 is approximately 0.1436.
Explanation:
To find the conditional probability P(X>120|X>105) , we first need to understand what this probability represents.
1. P(X>120|X>105) is the probability that the serum zinc level is greater than 120 given that it is already known to be greater than 105.
Given that X is normally distributed with mean = 90 and standard deviation SD = 15 , we can use the properties of a normal distribution to calculate this probability.
2. First, we calculate the Z-scores for the values 105 and 120 using the formula:
[tex]\( Z = \frac{X - \mu}{\sigma} \)[/tex]
- For X = 105: [tex]\( Z_{105} = \frac{105 - 90}{15} = 1 \)[/tex]
- For X = 120: [tex]\( Z_{120} = \frac{120 - 90}{15} = 2 \)[/tex]
3. Next, we find the area to the right of each Z-score using a standard normal distribution table or calculator:
- For Z = 1, the area to the right is approximately 0.1587.
- For Z = 2, the area to the right is approximately 0.0228.
4. To find the conditional probability, we use the formula for conditional probability:
[tex]\( Z_{120} = \frac{120 - 90}{15} = 2 \)[/tex]
5. In this case, event A is X > 120 and event B is X > 105. The intersection of events A and B is X > 120 since if X is greater than 120, it is also greater than 105.
6. Therefore, P(X>120|X>105) = [tex]\frac{P(X > 120)}{P(X > 105)} \)[/tex][tex]\ \ \( = \frac{0.0228}{0.1587} \approx 0.1436 \)[/tex]
Therefore, the conditional probability P(X>120|X>105) of the serum zinc level being greater than 120 given that it is already known to be greater than 105 is approximately 0.1436.
Your question is incomplete, but most probably the complete question was:
The serum zinc level X in micrograms per deciliter for males between ages 15 and 17 has a distribution that is approximately normal with mean=90 and SD=15.
What is the conditional probability P(X>120|X>105) of the serum zinc level x in micrograms per deciliter?
