he lengths of pregnancies are normally distributed with a mean of 271 days and a standard deviation of 25 days. If 100 women are randomly selected, find the probability that they have a mean pregnancy between 271 days and 273 days. ***
he lengths of pregnancies are normally distributed with a mean of 271 days and a standard deviation of 25 days. If 100 women are randomly selected, find the probability that they have a mean pregnancy between 271 days and 273 days. ***
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
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![**Probability Calculation for Mean Pregnancy Length**
The lengths of pregnancies are normally distributed with a mean of 271 days and a standard deviation of 25 days. If 100 women are randomly selected, find the probability that they have a mean pregnancy between 271 days and 273 days.
**Options:**
A. 0.5517
B. 0.2119
C. 0.2881
D. 0.7881
To solve this, we can use the properties of the normal distribution. The first step is to standardize the values and find the corresponding probabilities from the z-table.
To do this, calculate the Z-scores for 273 and 271 days and then find the area under the standard normal curve between these two Z-scores.
To standardize the values:
\[ Z = \frac{(X - \mu)}{(\sigma / \sqrt{n})} \]
where:
- \(X\) is the value of interest,
- \(\mu\) is the mean,
- \(\sigma\) is the standard deviation,
- \(n\) is the number of samples.
For 271 days:
\[ Z_{271} = \frac{(271 - 271)}{(25 / \sqrt{100})} = \frac{0}{2.5} = 0 \]
For 273 days:
\[ Z_{273} = \frac{(273 - 271)}{(25 / \sqrt{100})} = \frac{2}{2.5} = 0.8 \]
Now, we need to find the area under the normal curve between Z = 0 and Z = 0.8. Using a Z-table:
- Probability(Z < 0) = 0.5
- Probability(Z < 0.8) ≈ 0.7881
The probability of having a mean pregnancy length between 271 and 273 days is:
\[ P(0 < Z < 0.8) = 0.7881 - 0.5 = 0.2881 \]
Thus, the closest option is:
C. 0.2881](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2a348127-0366-4d6a-8a8b-87b3575e425d%2F24187142-42bc-4e65-90c0-69a10ecff36e%2F249kihr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Probability Calculation for Mean Pregnancy Length**
The lengths of pregnancies are normally distributed with a mean of 271 days and a standard deviation of 25 days. If 100 women are randomly selected, find the probability that they have a mean pregnancy between 271 days and 273 days.
**Options:**
A. 0.5517
B. 0.2119
C. 0.2881
D. 0.7881
To solve this, we can use the properties of the normal distribution. The first step is to standardize the values and find the corresponding probabilities from the z-table.
To do this, calculate the Z-scores for 273 and 271 days and then find the area under the standard normal curve between these two Z-scores.
To standardize the values:
\[ Z = \frac{(X - \mu)}{(\sigma / \sqrt{n})} \]
where:
- \(X\) is the value of interest,
- \(\mu\) is the mean,
- \(\sigma\) is the standard deviation,
- \(n\) is the number of samples.
For 271 days:
\[ Z_{271} = \frac{(271 - 271)}{(25 / \sqrt{100})} = \frac{0}{2.5} = 0 \]
For 273 days:
\[ Z_{273} = \frac{(273 - 271)}{(25 / \sqrt{100})} = \frac{2}{2.5} = 0.8 \]
Now, we need to find the area under the normal curve between Z = 0 and Z = 0.8. Using a Z-table:
- Probability(Z < 0) = 0.5
- Probability(Z < 0.8) ≈ 0.7881
The probability of having a mean pregnancy length between 271 and 273 days is:
\[ P(0 < Z < 0.8) = 0.7881 - 0.5 = 0.2881 \]
Thus, the closest option is:
C. 0.2881
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