Can someone help me figure out b. and c.? I understand a. bu the others are confusing for me to understand.
The mean amount of time it takes a kidney stone to pass is 13 days and the standard deviation is 4 days. Suppose that one individual is randomly chosen. Let X = time to pass the kidney stone. Round all answers to 4 decimal places where possible.
a. What is the distribution of X? X ~ N(13 , 4)
b. Find the probability that a randomly selected person with a kidney stone will take longer than 15 days to pass it.
c. Find the minimum number for the upper quarter of the time to pass a kidney stone.
days.
To answer b. and c., we can use the standard normal distribution table or a statistical calculator to find the probabilities and values. Here's how you can solve both b. and c. step by step:
b. Find the probability that a randomly selected person with a kidney stone will take longer than 15 days to pass it.
1. Convert the problem into a standard normal distribution:
Since we know the mean (μ) and standard deviation (σ) of the original distribution (X ~ N(13, 4)), we can convert it to a standard normal distribution (Z) by using the formula:
Z = (X - μ) / σ
In this case, X = 15, μ = 13, and σ = 4. Plugging in these values, we get:
Z = (15 - 13) / 4 = 2 / 4 = 0.5
2. Find the probability using the standard normal distribution table:
Look up the Z-value (0.5) in the standard normal distribution table (also known as the Z-table) and find the corresponding probability. Let's call the probability P(Z > 0.5).
According to the Z-table, the P(Z > 0.5) is equal to approximately 0.3085.
Therefore, the probability that a randomly selected person will take longer than 15 days to pass the kidney stone is approximately 0.3085.
c. Find the minimum number for the upper quarter of the time to pass a kidney stone.
1. Convert the upper quarter into a Z-value:
The upper quarter, also known as the upper quartile, represents the 75th percentile of the distribution. To find the minimum number for the upper quarter, we need to find the corresponding Z-value for the 75th percentile.
In the standard normal distribution, the Z-value corresponding to the 75th percentile is approximately 0.6745. You can find this value directly from the Z-table or calculate it using statistical software.
2. Convert the Z-value back to the original distribution:
Now we can use the Z-value to convert back into the original distribution by using the formula:
X = Z * σ + μ
Plugging in the values, we get:
X = 0.6745 * 4 + 13 = 2.698 + 13 ≈ 15.698
Therefore, the minimum number for the upper quarter of the time to pass a kidney stone is approximately 15.698 days (rounded to 4 decimal places).
I hope this helps you understand how to solve b. and c. for the given problem. Let me know if you have any further questions.