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 10 and standard deviation 3.

Short answer

The interval which is expected to contain about 95% of the data values is [4,16].

Step-by-step solution

Identification of the Given Parameters

It's given that a bell-shaped distribution with mean \( \mu = 10 \) and standard deviation \( \sigma = 3 \). This set of parameters will form our basis for calculating the confidence interval.

Apply the 95% Rule

The 95% rule states that approximately 95% of the data values should be within 2 standard deviations of the mean in a bell-shaped and symmetric distribution. So, for this exercise, it would mean calculating a range that starts at mean - (2*standard deviation) and ends at mean + (2*standard deviation). In other words, the interval would be given by \(10 - 2*3\) and \(10 + 2*3\).

Calculation of the Interval

So, the lower limit of the interval is \(10 - 2*3 = 4\) and the upper limit of the interval is \(10 + 2*3 =16\). Therefore, about 95% of the data values are expected to fall in the range [4,16].

Key concepts

Bell-Shaped Distribution

Understanding the bell-shaped distribution is essential when analyzing data in many fields, as it represents a pattern often found in nature and human-made processes. This type of distribution, also known as the normal distribution or Gaussian distribution, appears as a symmetric, bell-shaped curve when graphed.

The hallmark of the bell-shaped distribution is its symmetry, which implies that most data points cluster around a central peak—the mean or average value—and fewer points are found as you move away from the center. When you see this distribution, you can expect that values much higher or much lower than the mean are relatively rare.

Distinguishing Features

Key characteristics include the mean, median, and mode being at the highest peak and the same value. Furthermore, the data tails off symmetrically on either side, decreasing in frequency as you move away from the mean. It's this shape that underlies many statistical rules and principles, such as the 95% rule used for creating confidence intervals.

Standard Deviation

When working with data, it's crucial to understand the level of variability or spread within the set. This is where standard deviation (\( \text{SD} \text{ or }\text{ \text{ sigma} \)), a key statistical tool, comes into play.

In essence, standard deviation measures how much the values in a data set vary from the average (mean) value. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation suggests that the data points are spread out over a wider range.

Practical Implication

In our exercise, the standard deviation is 3. This tells us that individual data points are, on average, 3 units away from the mean (in this case, 10). This measurement of dispersion is central to creating confidence intervals and understanding the reliability of the mean as a representative number for the entire data set.

95% Rule

The 95% rule is a shortcut that applies specifically to bell-shaped distributions and is particularly useful for quickly estimating the range in which most of the data falls. According to this rule, approximately 95% of the data values lie within two standard deviations of the mean—both above and below it.

This rule provides a way to build what is known as a confidence interval, a range that is likely to contain the true population parameter with a certain degree of certainty—in this case, 95%.

Application Scenario

Referring back to our exercise, with a mean of 10 and a standard deviation of 3, we apply the 95% rule. By doubling the standard deviation (3) and adding and subtracting it from the mean (10), we get the interval [4, 16], where we can say with 95% confidence that most of the data values will fall.