A sample data set with a bell-shaped distribution and size...

A sample data set with a bell-shaped distribution and size $n=128$ has mean $x^{\wedge}=2$ and standard deviation $s=1.1$. Find the approximate number of observations in the data set that lie:
a. below-0.2;
b. below 3.1;
c. between -1.3 and 0.9 .

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric around its mean, displaying a bell-shaped curve. It is fundamental in statistics because many variables tend to follow this distribution, and it underpins many inferential techniques, allowing for probabilities to be associated with specific intervals around the mean.

Z-Score Standardization

Z-score standardization is the process of transforming a value from a normal distribution into a standardized score by subtracting the mean and then dividing by the standard deviation. This conversion allows different normal distributions to be compared using the standard normal distribution, facilitating the use of standard tables and probability calculations.

Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 rule, provides a quick estimate of the proportion of data that falls within one, two, and three standard deviations from the mean in a normal distribution. This rule is useful for making approximate probability and frequency predictions without having to perform extensive calculations.

Area Under the Curve

In the context of a normal distribution, the area under the curve represents the probability that a random observation lies within a specified interval. By determining the area, one can approximate the number of observations in a sample that meet particular criteria, bridging the gap between probability theory and real-world data counts.

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