91Ó°ÊÓ

In the context of a normal distribution, **mean** (\( \text{\textmu} \)) and **standard deviation** (\( \text{\textsigma} \)) are fundamental concepts. The mean represents the average or central value of the data. For example, in our exercise, the mean length of human pregnancies is \( 266 \) days. This means that on average, pregnancies last \( 266 \) days.
The standard deviation, on the other hand, measures the spread or dispersion of the data around the mean. A smaller standard deviation indicates that the data points are closer to the mean, while a larger standard deviation shows more variability. In this exercise, the standard deviation is \( 16 \) days. This tells us that the lengths of pregnancies tend to vary by about \( 16 \) days on either side of the mean.
So, in summary, mean helps us understand the central tendency of the data, and standard deviation indicates how much the data tends to deviate from this central point.

When interpreting probabilities in the context of a normal distribution, imagine the entire area under the curve as representing 100% of all possible outcomes.
For part (a) of the exercise, the area to the right of \( x = 280 \) is 0.1908. This area tells us the probability of a pregnancy lasting more than \( 280 \) days. In other words, there is a 19.08% chance of a pregnancy going beyond \( 280 \) days.
In part (b), the area between \( x = 230 \) and \( x = 260 \) is 0.3416. This means there is a 34.16% chance that a pregnancy will last within this range. Another way to interpret this is that if we randomly select a pregnancy, there’s a 34.16% probability it will fall between \( 230 \) and \( 260 \) days.

The area under the normal distribution curve represents probabilities. Each section of the curve corresponds to a different probability. The curve is symmetrical and centered around the mean.
For example, the area to the right of \( x = 280 \) represents the probability of a pregnancy lasting more than \( 280 \) days, which is 0.1908 or 19.08%. This area is shaded to the right of \( x = 280 \).
Similarly, the area between \( x = 230 \) and \( x = 260 \) represents the probability of a pregnancy falling within that range, which is 0.3416 or 34.16%. This area is shaded between \( x = 230 \) and \( x = 260 \).
Understanding the area under the curve helps us interpret different probabilities and make sense of the data.

The normal distribution curve is a bell-shaped curve that is symmetrical around the mean. It is characterized by its mean \( \text{\textmu} \) and standard deviation \( \text{\textsigma} \).
The highest point on the curve represents the mean, where most of the data points are concentrated. As you move away from the mean, the curve drops off, showing fewer occurrences of the data.
In the context of our exercise, the normal distribution curve has a mean of \( 266 \) days and a standard deviation of \( 16 \) days. This curve allows us to visualize the distribution of pregnancy lengths.
Areas under the curve represent different probabilities, making it a powerful tool for statistical analysis. For instance, whether interpreting probabilities for a pregnancy lasting more than \( 280 \) days or falling between \( 230 \) and \( 260 \) days, the normal distribution curve helps us understand these probabilities effectively.