Question

On average, an american professional football game lasts about three hours, even though the ball is actually in play only 11 minutes ( , january 14, 2014). Let the standard deviation be 0.4 hour. a. Use Chebyshev's theorem to approximate the proportion of games that last between 2.2 hours and 3.8 hours. b. Assume a bell-shaped distribution to approximate the proportion of games that last between 2.2 hours and 3.8 hours.

Answer 1

To approximate the proportion of games that last between 2.2 hours and 3.8 hours, we can use Chebyshev's theorem or assume a bell-shaped distribution.

Step-by-step explanation:

To approximate the proportion of games that last between 2.2 hours and 3.8 hours using Chebyshev's theorem, we can calculate the range of values within a certain number of standard deviations from the mean. Chebyshev's theorem states that for any distribution, regardless of its shape, at least (1 - 1/z^2) of the data falls within z standard deviations from the mean, where z is any positive number greater than 1. The proportion of games that last between 2.2 hours and 3.8 hours can be approximated by finding the range of values within 2 standard deviations from the mean.

Using Chebyshev's theorem formula:

Proportion = 1 - 1/z^2

where z = (3.8 - mean) / standard deviation

By plugging in the given values, we can calculate the proportion of games that last between 2.2 hours and 3.8 hours.

To approximate the proportion of games that last between 2.2 hours and 3.8 hours assuming a bell-shaped distribution, we can use the properties of the normal distribution. The proportion can be calculated by finding the area under the normal curve between 2.2 hours and 3.8 hours, using the mean and standard deviation.

We can use the cumulative distribution function (CDF) of a normal distribution to find the proportion of games that last between 2.2 hours and 3.8 hours.