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 200 and standard deviation 25.

Short answer

The interval expected to contain about 95% of the data values is [150, 250]

Step-by-step solution

Identify the Mean and Standard Deviation

The mean (μ) of the distribution is 200 and the standard deviation (σ) is 25.

Apply the Empirical Rule

According to the Empirical Rule (also known as the 68-95-99.7 rule), for a normal distribution, about 95% of the data is within two standard deviations of the mean. So to get the interval, we subtract and add two times the standard deviation from the mean.

Calculate the Interval Boundaries

To find the lower boundary, we subtract two times the standard deviation from the mean, so \(200 - 2*25 = 150\). To find the upper boundary, we add two times the standard deviation to the mean, so \(200 + 2*25 = 250\). This gives us the interval [150, 250]

State the Final Answer

The interval that is expected to contain about 95% of the data values for this distribution is [150, 250]

Key concepts

Understanding the Normal Distribution

Imagine a bell-shaped curve where most of the values are concentrated around a central point, gently tapering off towards the ends; this is what a normal distribution looks like. It's a continuous probability distribution that is symmetric about the mean, which means that the data is equally distributed on both sides of the mean.

Mathematically, the normal distribution is described by two parameters: the mean (average) and the standard deviation (a measure of spread). In real-world terms, think of the mean as where the peak of the bell curve sits on the horizontal axis, representing the most common value. The standard deviation then tells us how much the values tend to deviate from the mean on average.

A normal distribution has some fascinating properties. For example, about 68% of the data will fall within one standard deviation of the mean, about 95% within two standard deviations, and about 99.7% within three standard deviations. This predictable pattern is what makes the normal distribution so useful in statistics for making predictions and understanding data.

Deciphering Standard Deviation

To truly grasp the empirical rule and the concept of confidence intervals, it's crucial to understand what standard deviation is. Standard deviation is a statistical measure of the amount of variability or spread in a set of data. In simpler terms, it's an indication of how far the data points are from the average value (mean).

Imagine you have scores from a test. If everyone scored very close to the class average, we would say the standard deviation is small. Conversely, if students’ scores were all over the place, this would mean a large standard deviation. To calculate it, we sum up all the squared differences from the mean, divide by the number of data points, and take the square root. In the context of the normal distribution, a small standard deviation means the bell curve is steep and narrow, while a large one makes it shallow and wide.

In the example exercise, a standard deviation of 25 means that most data points fall within 25 units of the mean, either above or below. Remember, the standard deviation is a key component in determining the spread of a normal distribution and is crucial when using the empirical rule to calculate confidence intervals.

The 95% Confidence Interval in Practice

The 95% confidence interval is a range of values that you can be 95% certain contains the true mean of the population. It's a commonly used concept in statistics as it provides a way to estimate the reliability of sample data. In the context of the normal distribution and the empirical rule, using the 95% confidence interval means we can predict that if we picked data points at random from our distribution, 95% of the time, those points would fall within this range.

The empirical rule makes it easy to calculate the 95% confidence interval for data that is normally distributed. We simply take the mean and add and subtract two standard deviations to find the two boundaries. In our exercise, with a mean of 200 and a standard deviation of 25, we end up with a 95% confidence interval between 150 and 250. This interval is what statisticians use when they want to state that they are 95% sure the true mean value of their data falls within a specified range.