Math Problem Statement
Assume that the duration of human pregnancies can be described by a Normal model with mean 267 days and standard deviation 16 days. a) What percentage of pregnancies should last between 270 and 282 days? b) At least how many days should the longest 15% of all pregnancies last? c) Suppose a certain obstetrician is currently providing prenatal care to 46 pregnant patients. Let y overbar represent the mean length of their pregnancies. According to the Central Limit Theorem, what is the distribution of this sample mean, y overbar? Specify the model, mean, and standard deviation. d) What's the probability that the mean duration of these patients' pregnancies will be less than 259 days
Solution
Let's solve each part step by step:
Part (a): Percentage of pregnancies lasting between 270 and 282 days
We are tasked with finding the percentage of pregnancies lasting between 270 and 282 days. The model is a normal distribution with:
- Mean () = 267 days
- Standard deviation () = 16 days
The corresponding -scores are calculated as:
For :
For :
Using standard normal distribution tables or a calculator, the cumulative probabilities are:
The percentage of pregnancies lasting between 270 and 282 days is:
Thus, 25.11% of pregnancies last between 270 and 282 days.
Part (b): Longest 15% of all pregnancies
We need to find the value of such that the top 15% of pregnancies have lengths greater than . The corresponding cumulative probability is:
From standard normal tables or a calculator, the -score corresponding to is approximately:
Converting this -score back to the original scale:
Thus, the longest 15% of pregnancies last at least 283.6 days.
Part (c): Distribution of the sample mean ()
According to the Central Limit Theorem, the sample mean () is normally distributed with:
- Mean: days
- Standard deviation:
Thus, the distribution of is:
Part (d): Probability that the mean duration is less than 259 days
We are tasked with finding , where:
The corresponding -score is:
Using standard normal tables or a calculator:
Thus, the probability that the mean duration of these patients' pregnancies is less than 259 days is approximately 0.035%.
Summary of Results
- (a)
- (b) Longest 15% of pregnancies last at least 283.6 days.
- (c)
- (d) Probability is approximately 0.035%.
Do you have any questions or want a more detailed explanation for any part?
Follow-Up Questions
- How is the -score used to convert between raw scores and probabilities in a normal distribution?
- Why does the Central Limit Theorem apply when dealing with sample means?
- Can you explain how the standard deviation of the sample mean decreases as the sample size increases?
- How would the results change if the sample size in part (c) were doubled?
- What are some real-world applications of normal distribution in medicine?
Tip
When using the Central Limit Theorem, always verify that the sample size is large enough for the theorem to apply effectively (usually ).
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Math Problem Analysis
Mathematical Concepts
Normal Distribution
Z-scores
Probability
Central Limit Theorem
Sampling Distribution
Formulas
z = (x - μ) / σ
P(Z ≤ z) for cumulative probability
σ_sample = σ / √n
Theorems
Central Limit Theorem
Suitable Grade Level
Grades 11-12, Undergraduate Statistics
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