The length of human pregnancies is approximately normal with mean mu = 266 days and standard deviation sigma = 16 days. Complete parts (a) through (f).
(a) What is the probability that a randomly selected pregnancy lasts less than 261 days?
The probability that a randomly selected pregnancy lasts less than 261 days is approximately
(Round to four decimal places as needed.)
The Correct Answer and Explanation is :
To determine the probability that a randomly selected pregnancy lasts less than 261 days, given that human pregnancy lengths follow a normal distribution with a mean (μ) of 266 days and a standard deviation (σ) of 16 days, we can follow these steps:
1. Standardize the Value to a Z-Score:
First, we convert the raw score (261 days) to a standard normal variable (z-score) using the formula:
[ z = \frac{X – \mu}{\sigma} ]
Where:
- ( X = 261 ) days
- ( \mu = 266 ) days
- ( \sigma = 16 ) days
Calculating the z-score:
[ z = \frac{261 – 266}{16} = \frac{-5}{16} \approx -0.3125 ]
2. Find the Corresponding Probability:
The z-score of approximately -0.3125 indicates how many standard deviations the value of 261 days is below the mean. To determine the probability associated with this z-score, we can use standard normal distribution tables or computational tools.
Using a standard normal distribution calculator:
- Mean (μ): 266 days
- Standard Deviation (σ): 16 days
- Value (X): 261 days
Inputting these values, the calculator provides a cumulative probability of approximately 0.3770.
3. Interpret the Result:
A cumulative probability of 0.3770 means that there is a 37.70% chance that a randomly selected pregnancy will last less than 261 days.
Explanation:
In a normal distribution, the mean represents the central tendency of the data, and the standard deviation measures the spread or variability around the mean. Approximately 68% of values lie within one standard deviation of the mean (between μ – σ and μ + σ), 95% within two standard deviations, and 99.7% within three standard deviations—a concept known as the empirical rule. In this context, a pregnancy length of 261 days is slightly below the mean, falling within the first standard deviation but closer to the lower end.
Visual Representation:
To visualize this, consider a normal distribution curve centered at 266 days. A value of 261 days lies to the left of the mean, and the area under the curve to the left of this value represents the cumulative probability of approximately 37.70%.
Conclusion:
By standardizing the value and referring to the standard normal distribution, we determined that there is a 37.70% probability that a randomly selected pregnancy will last less than 261 days. This method leverages the properties of the normal distribution to assess probabilities associated with specific values.