Question

Use the \(95 \%\) rule and the fact that the summary statistics come from a distribution that is symmetric and bell-shaped to find an interval that is expected to contain about \(95 \%\) of the data values. A bell-shaped distribution with mean 1000 and standard deviation 10.

Short answer

The interval that is expected to contain about 95% of the data values goes from 980 to 1020.

Step-by-step solution

Understand the problem

We are looking for an interval that contains 95% of the data points for a normal distribution. Given the mean \( \mu = 1000 \) and standard deviation \( \sigma = 10 \) of the distribution, we use the 95% rule which states that for a normal distribution, about 95% of the data comes within two standard deviations of the mean.

Apply the 95% Rule

To apply this rule, we calculate the boundaries of the interval by subtracting and adding two times the standard deviation from the mean. The lower boundary of the interval is given by \( \mu - 2\sigma \), and the upper boundary of the interval is given by \( \mu + 2\sigma \).

Calculate the Boundaries

We substitute the given values into the expressions for the lower and upper boundaries. The lower boundary is \(1000 - 2(10) = 980\) and the upper boundary is \(1000 + 2(10) = 1020\). Therefore, the interval which contains about 95% of the data values is from 980 to 1020.

Key concepts

Understanding Normal Distribution

The concept of normal distribution is a foundational pillar in statistics and various scientific disciplines. It is often represented as a symmetric, bell-shaped curve where the majority of data points cluster around a central point known as the mean. The normal distribution is paramount because it naturally arises in countless real-world scenarios, from heights of people to measurement errors, thanks to the Central Limit Theorem.

Envision the situation where multiple factors with random variations affect a particular outcome. Even if we don’t know precisely how these factors individually operate, when aggregated, they tend to form a distribution of results that looks like the classic 'bell curve'. For example, if we measure the heights of a large group of people, although each individual's height is determined by many different genetic and environmental factors, the overall distribution of heights tends to be normally distributed.

Moreover, since normal distributions follow a predictable pattern, rules like the 95% rule emerge, allowing us to infer that certain proportions of data will lie within specific intervals relative to the mean.

The Role of Standard Deviation

When talking about normal distributions, the term standard deviation frequently comes up. It's a measure of variability or dispersion, indicating how much the individual data points in a dataset deviate from the mean value. In the context of a bell-shaped curve, it signifies how spread out the data is. If the standard deviation is small, it means the data points are closely packed around the mean, while a large standard deviation observes a more scattered dataset.

In practical applications, standard deviation contributes to determining confidence intervals and predicting outcomes. For instance, in the exercise, knowing that the standard deviation is 10 allows us to apply the 95% rule to estimate that most of the data values (95% of them) will be within two standard deviations from the mean. This ability to predict where data points are likely to fall is crucial in fields as diverse as financial risk assessment, quality control, and even sports analytics. The calculated interval from the exercise (980 to 1020) translates to real-world expectations that nearly all measurements will fall within this range, giving us a powerful insight into the data's behavior.

Insights into Bell-Shaped Distribution

A bell-shaped distribution is another name for the normal distribution because of its graphical representation that resembles a bell. This distribution is symmetrical, which means the left and right sides of the curve are mirror images of each other concerning the central peak, where the mean lies. The tails of the curve stretch towards infinity, implying that while most data are close to the mean, a few data points can still be far off.

Understanding the concept of a bell-shaped distribution is crucial because it provides a framework for making inferences about data sets. By recognizing the shape of the distribution, even without every single data point, we can draw conclusions about the likelihood of certain outcomes—for example, the probability of a value falling within a particular range. This holds especially true when referring to the so-called 'empirical rule' which states that approximately 68%, 95%, and 99.7% of data in a normal distribution falls within one, two, and three standard deviations from the mean, respectively. This rule is the foundation of the exercise's solution, showcasing this distribution's predictive power.