the pregnancy length in days for a population of new mothers can be approximated by a normal distribution with a mean of 270 days and a standard deviation of 12 days.
a. what is the minimum pregnancy length that can be in the top 11% of pregnancy lengths?
b. What is the maximum pregnancy length that can be in the bottom 6% of pregnancy lengths?
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability and its Z score. Insert data into equation below to solve for score (length).
Z = (score-mean)/SD
a. Well, if we're talking about the top 11% of pregnancy lengths, we're basically looking for the minimum outlier here. So, let me put on my mathematician clown wig and calculate that for you.
To find the minimum pregnancy length in the top 11%, we need to find the z-score corresponding to the 11th percentile. Luckily, the z-score formula is here to save the day: z = (x - μ) / σ, where x is the value we're looking for, μ is the mean, and σ is the standard deviation.
Using this formula, we can rewrite it as x = (z * σ) + μ.
So, let's calculate the z-score corresponding to the 11th percentile: z = invNorm(0.11) (invNorm refers to the inverse normal distribution).
Plugging in the values, we get x = (invNorm(0.11) * 12) + 270.
Unfortunately, as a clown bot, I don't have access to my trusty calculator right now. But don't worry, you can find it using a calculator or software that performs inverse normal distribution calculations. Just don't forget to substitute invNorm(0.11) for its actual value!
b. Ah, now we're looking for the maximum pregnancy length in the bottom 6%. Let's flip the script!
To find the maximum pregnancy length in the bottom 6%, we need to find the z-score corresponding to the 6th percentile. Following the same formula as before, we have x = (z * σ) + μ.
Now, we can calculate the z-score corresponding to the 6th percentile: z = invNorm(0.06).
Substituting the values, we get x = (invNorm(0.06) * 12) + 270.
Of course, you'll need to find the value of invNorm(0.06) using an appropriate tool.
Remember, it's always good to double-check my calculations using the actual values for invNorm! Keep in mind that I'm just a clown bot trying to provide some laughter along with mathematical guidance.
To find the minimum pregnancy length in the top 11% and the maximum pregnancy length in the bottom 6% using a normal distribution with a mean of 270 days and a standard deviation of 12 days, we can use the z-score formula.
a. To find the minimum pregnancy length in the top 11%, we need to find the z-score corresponding to the cutoff point. The z-score can be found using the standard normal distribution table or a calculator. The z-score corresponding to the top 11% is approximately 1.23.
Once we have the z-score, we can calculate the minimum pregnancy length using the formula:
Minimum pregnancy length = mean + (z-score * standard deviation)
Minimum pregnancy length = 270 + (1.23 * 12)
Minimum pregnancy length = 270 + 14.76
Minimum pregnancy length ≈ 284.76 days
Therefore, the minimum pregnancy length that can be in the top 11% is approximately 284.76 days.
b. To find the maximum pregnancy length in the bottom 6%, we need to find the z-score corresponding to the cutoff point. The z-score can be found using the standard normal distribution table or a calculator. The z-score corresponding to the bottom 6% is approximately -1.56.
Once we have the z-score, we can calculate the maximum pregnancy length using the formula:
Maximum pregnancy length = mean + (z-score * standard deviation)
Maximum pregnancy length = 270 + (-1.56 * 12)
Maximum pregnancy length = 270 - 18.72
Maximum pregnancy length ≈ 251.28 days
Therefore, the maximum pregnancy length that can be in the bottom 6% is approximately 251.28 days.
To find the minimum and maximum pregnancy lengths for specific percentiles, we can use the concept of Z-scores and the standard normal distribution.
The Z-score formula is given by:
Z = (X - μ) / σ
where:
Z is the Z-score
X is the observed value
μ is the mean
σ is the standard deviation
We want to find the Z-score that corresponds to the given percentiles (11% and 6%), and then use it to calculate the corresponding pregnancy lengths.
a. To find the minimum pregnancy length that can be in the top 11% of pregnancy lengths:
Step 1: Convert the given percentile to a Z-score. Since we want the top 11%, we need to find the Z-score that corresponds to the cumulative probability of 1 - 0.11 = 0.89. We can use a Z-table or a calculator to find this value. In this case, the Z-score is approximately 1.231.
Step 2: Use the Z-score formula to find the minimum pregnancy length:
1.231 = (X - 270) / 12
Rearrange the formula to solve for X:
X = (1.231 * 12) + 270
X ≈ 284.77
Therefore, the minimum pregnancy length that can be in the top 11% is approximately 284.77 days.
b. To find the maximum pregnancy length that can be in the bottom 6% of pregnancy lengths:
Step 1: Convert the given percentile to a Z-score. Since we want the bottom 6%, we need to find the Z-score that corresponds to the cumulative probability of 0.06. We can use a Z-table or a calculator to find this value. In this case, the Z-score is approximately -1.556.
Step 2: Use the Z-score formula to find the maximum pregnancy length:
-1.556 = (X - 270) / 12
Rearrange the formula to solve for X:
X = (-1.556 * 12) + 270
X ≈ 251.25
Therefore, the maximum pregnancy length that can be in the bottom 6% is approximately 251.25 days.