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: The intervals are (415, 425), (410, 430), and (405, 435). Approximately 68% of the measurements fall within the first interval, 95% within the second interval, and 99.7% within the third interval.

Step-by-step solution

Find the sample standard deviation

First, we need to find the sample standard deviation, which is the square root of the sample variance. The sample variance is given as 25, so the sample standard deviation is \(\sqrt{25} = 5\).

Calculate the intervals

Now we'll calculate the three intervals: 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 according to the Empirical Rule

According to the Empirical Rule: 1. Approximately 68% of the measurements fall within one standard deviation of the mean, i.e., between 415 and 425. 2. Approximately 95% of the measurements fall within two standard deviations of the mean, i.e., between 410 and 430. 3. Approximately 99.7% of the measurements fall within three standard deviations of the mean, i.e., between 405 and 435.