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 \(\bar{x} \pm 3 s\) and then state the approximate fractions of measurements you would expect to fall into these intervals according to the Empirical Rule.

Short answer

Answer: According to the Empirical Rule, the intervals of attendance are approximately (415, 425) for 68% of the data, (410, 430) for 95% of the data, and (405, 435) for 99.7% of the data.

Step-by-step solution

Calculate the standard deviation from variance

We are given the variance as \(25\). The standard deviation (s) is the square root of the variance: \(s = \sqrt{25}\) \(s = 5\)

Calculate the intervals \(\bar{x} \pm s, \bar{x} \pm 2 s,\) and \(\bar{x} \pm 3 s\)

The sample mean (\(\bar{x}\)) is given as 420. We will now calculate the three intervals using the standard deviation: 1. \(\bar{x} \pm s = 420 \pm 5 = (415, 425)\) 2. \(\bar{x} \pm 2s = 420 \pm 2(5) = (410, 430)\) 3. \(\bar{x} \pm 3s = 420 \pm 3(5) = (405, 435)\)

State the approximate fractions of measurements according to the Empirical Rule

Now that we have the three intervals, we can state the approximate fractions of measurements: 1. 68% of the data is expected to fall within the interval \(\bar{x} \pm s = (415, 425)\) 2. 95% of the data is expected to fall within the interval \(\bar{x} \pm 2s = (410, 430)\) 3. 99.7% of the data is expected to fall within the interval \(\bar{x} \pm 3s = (405, 435)\) In conclusion, according to the Empirical Rule, we can expect approximately 68% of the attendances to fall within the interval (415, 425), 95% within the interval (410, 430), and 99.7% within the interval (405, 435).

Key concepts

Understanding Standard Deviation

Standard deviation is a key concept in statistics that helps us understand how much variation or dispersion exists in a set of data. It tells us how much individual data points deviate, on average, from the mean of the dataset. In simpler terms, it measures the extent to which numbers are spread out. - A small standard deviation means the data points tend to be close to the mean. - A large standard deviation means there is more spread among the data points. In our example, the standard deviation is calculated from the variance. Since the variance is given as 25, we take the square root of 25 to find the standard deviation, resulting in a value of 5. Simply put, the average deviation of attendance numbers from the mean (420) is 5.

Exploring Variance

Variance is a measure of how far each number in the dataset is from the mean and, consequently, from every other number in the dataset. It is essentially the average of the squared differences from the mean, providing a sense of how much the numbers in a dataset are spread out.In mathematical terms, variance is calculated as:\[ \sigma^2 = \frac{1}{N} \sum_{i=1}^{N}(x_i - \bar{x})^2 \]where \( \sigma^2 \) represents variance, \( x_i \) are individual data points, and \( \bar{x} \) is the mean.In our scenario with the basketball game attendances, the variance is provided as 25. This value helps us determine the standard deviation by taking its square root. Understanding variance is crucial because it underpins the calculation of standard deviation.

Normal Distribution

A normal 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. In data visualization, it is often represented as a bell curve. The normal distribution is fundamental because it describes how data is naturally distributed in many real-world situations.Key attributes of a normal distribution: - The mean, median, and mode are all equal. - The bell curve is symmetric at the center (around the mean, \( \bar{x} \)). - Approximately 68% of the data falls within one standard deviation from the mean. - 95% falls within two standard deviations, and 99.7% lies within three.Given our attendance data, it’s assumed to be normally distributed, which allows us to apply the Empirical Rule to determine the fraction of data within each interval.

Confidence Intervals and the Empirical Rule

Confidence intervals provide a range of values which is likely to contain a population parameter, like the mean, with a certain level of confidence. When applied to a normal distribution, these intervals help us understand the range in which most data points are expected to fall.The Empirical Rule, also known as the 68-95-99.7 rule, is closely related to confidence intervals. It gives us a way to estimate the probability of data falling within certain standard deviation ranges:- About 68% of data falls within one standard deviation (\( \bar{x} \pm s \))- Roughly 95% of data falls within two (\( \bar{x} \pm 2s \))- Nearly 99.7% of data falls within three standard deviations (\( \bar{x} \pm 3s \))In the context of our exercise, the intervals calculated using this rule help provide confidence about where most basketball game attendance figures are expected to fall in relation to the mean.