Question

Attendances at a high school's basketball games were recorded and found to have a sample mean and variance of 420 and \(25,\) respectively. Calculate \(\bar{x} \pm s, \bar{x} \pm 2 s,\) and \(\bar{x} \pm 3 s .\) What fraction of measurements would you expect to fall into these intervals according to the Empirical Rule?

Short answer

Question: Calculate the intervals \(\bar{x} \pm s\), \(\bar{x} \pm 2s\), and \(\bar{x} \pm 3s\) and state the fraction of measurements expected to fall into these intervals according to the Empirical Rule for the given sample mean and variance of attendances at basketball games. The sample mean is 420 and the variance is 25. Answer: The intervals are as follows: 1. \(\bar{x} \pm s = (415, 425)\) with approximately 68% of measurements. 2. \(\bar{x} \pm 2s = (410, 430)\) with approximately 95% of measurements. 3. \(\bar{x} \pm 3s = (405, 435)\) with approximately 99.7% of measurements.

Step-by-step solution

Find the standard deviation from the given variance

The variance is given as 25, and the standard deviation is the square root of the variance. Therefore, we can calculate the standard deviation \(s\) as follows: $$ s = \sqrt{25} = 5 $$

Calculate \(\bar{x} \pm s\)

We are given the sample mean \(\bar{x} = 420\) and the standard deviation \(s = 5\). Therefore, the interval \(\bar{x} \pm s\) is: $$ 420 \pm 5 = (415, 425) $$

Calculate \(\bar{x} \pm 2s\)

We have the sample mean \(\bar{x} = 420\) and the standard deviation \(s = 5\). Therefore, the interval \(\bar{x} \pm 2s\) is: $$ 420 \pm 2 \cdot 5 = (410, 430) $$

Calculate \(\bar{x} \pm 3s\)

We have the sample mean \(\bar{x} = 420\) and the standard deviation \(s = 5\). Therefore, the interval \(\bar{x} \pm 3s\) is: $$ 420 \pm 3 \cdot 5 = (405, 435) $$

State the fraction of measurements expected in each interval according to the Empirical Rule

According to the Empirical Rule, for a normal distribution, the fraction of measurements falling into each interval is as follows: - \(\bar{x} \pm s\): Approximately 68% of measurements fall into this interval. - \(\bar{x} \pm 2s\): Approximately 95% of measurements fall into this interval. - \(\bar{x} \pm 3s\): Approximately 99.7% of measurements fall into this interval.

Key concepts

Standard Deviation

When analyzing data, it's important to have a measure of the spread or variability of the set. Standard deviation (denoted as s) is one of the most common indicators of this variability. It tells us how much the individual data points differ, on average, from the mean of the dataset.

To compute the standard deviation, you must first find the square root of the variance. The variance represents the average squared deviation from the mean and is usually denoted as s². So, for our basketball games attendance example, given a variance of 25, the standard deviation is calculated as \( s = \sqrt{25} = 5 \). This value of 5 is a unit of measure that describes that, on average, the attendance numbers deviate from the mean by 5 attendees.

Understanding the standard deviation aids in interpreting the spread of data around the mean, helping us predict the distribution of attendance for future games and enabling a more accurate analysis of the data.

Sample Mean

The sample mean, often represented by \( \bar{x} \), is the average value in a sample set of data. It is determined by adding all the numbers in the dataset and dividing by the total number of observations. In our high school basketball game example, the sample mean is given as 420. This signifies that, on average, 420 individuals attend the games.

The mean is a central position around which other statistics are often framed and compared, such as the intervals defined by adding or subtracting multiples of the standard deviation. For instance, calculating \( \bar{x} \pm s \) determines the range within which approximately 68% of the game attendances will fall if the data is normally distributed. Such calculations are pivotal in statistics as they help in making predictions and setting expectations for phenomena being observed.

Normal Distribution

Normal distribution, also known as the bell curve or Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

According to the Empirical Rule (also called the 68-95-99.7 rule), in a normal distribution, about 68% of values fall within one standard deviation (\( s \) away from the mean), about 95% of the values fall within two standard deviations (\( 2s \) away from the mean), and about 99.7% fall within three standard deviations (\( 3s \) away from the mean).

This rule is incredibly useful because it allows for a quick estimate of the spread of data points without intricate calculations. In the given exercise, by applying these percentages to our calculated intervals (\( \bar{x} \pm s \), \( \bar{x} \pm 2s \) and \( \bar{x} \pm 3s \) respectively), we can reasonably predict that most of the attendance figures would land within the ranges of 415-425, 410-430, and 405-435 respectively.