Question

A sample of heights of trees follows a normal distribution. In this sample, 68% of height measurements are expected to fall in the interval: \[ \text{mean} \pm \underline{\hspace{1cm}} \text{ standard deviation.} \] (Round off to the nearest integer.)

Hint:

In a normal distribution, 68% of the data lies within one standard deviation from the mean. This is a key property when analyzing data distributions.

Answer 1

Correct Answer: 0.99

In a normal distribution, approximately 68% of all values lie within one standard deviation from the mean. This is a well-known property of the normal distribution curve.

Step 1: Understand the concept of 68% range.
For any normally distributed dataset, the interval covering 68% of the data will be within one standard deviation on either side of the mean. This means the range is defined by: \[ \text{mean} \pm 1 \times \text{standard deviation.} \]

Step 2: Applying the formula.
The data provided indicates that 68% of the values fall within this range. So, to express the interval mathematically, we use: \[ \text{Interval} = \text{mean} \pm 1 \times \sigma \] where \( \sigma \) is the standard deviation of the dataset. This allows us to understand the variability or spread of data around the mean.

Step 3: Conclusion.
Thus, the interval containing 68% of the data points will be from \( \text{mean} - \sigma \) to \( \text{mean} + \sigma \), and the range is simply the standard deviation from the mean.