Final answer:
Using the empirical rule, we can estimate the length of the game at the 70th percentile to be approximately 135.24 minutes.
Step-by-step explanation:
The length of one basketball game, including rests and timeouts, follows a normal distribution with a mean of 130 minutes and a standard deviation of 10 minutes. We are asked to estimate the length of a randomly selected basketball game that is at the 70th percentile using the empirical rule (68-95-99.7).
The empirical rule states that approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.
Since the game length is normally distributed, we can use the z-score formula to determine how many standard deviations the 70th percentile is from the mean. The formula is:
z = (x - µ) / σ
where z is the z-score, x is the value we are interested in, µ is the mean, and σ is the standard deviation.
Since we are interested in finding the length of the game at the 70th percentile, we can calculate the z-score as follows:
z = (x - 130) / 10
For the 70th percentile, the z-score is approximately 0.524. To find the corresponding length of the game, we can rearrange the z-score formula and solve for x:
x = z * σ + µ
Substituting the values:
x = 0.524 * 10 + 130
x = 5.24 + 130
x ≈ 135.24
Therefore, we can estimate that the length of a randomly selected basketball game at the 70th percentile is approximately 135.24 minutes.