Question

At the U.S. Open Tennis Championship a statistician keeps track of every serve that a player hits during the tournament. The statistician reported that the mean serve speed of a particular player was 100 miles per hourâ (mph) and the standard deviation of the serve speeds was 13 mph. If nothing is known about the shape of theâ distribution, give an interval that will contain the speeds of at least eight-ninths of theâ player's serves.

a. 74 mph to 126 mph
b. 61 mph to 139 mph
c. 48 mph to 152 mph
d. 139 mph to 178 mph

Answer 1

b. 61 mph to 139 mph

Explanation:

Chebyshev's theorem states that

For k> 1

1 - 1/k²

Where:

At least 75% or 3/4 of data falls within 2 standard deviations from the mean - between μ – 2σ and μ + 2σ.

At least 88.89% of 8/9 of data falls within 3 standard deviations from the mean - between μ - 3σ and μ + 3σ.

From the question, we are told to find an interval that will contain the speeds of at least eight-ninths of the player's serves.

Mean (μ)= 100mph

Standard deviation (σ) = 13mph

At least 88.89% of 8/9 of data falls within 3 standard deviations from the mean - between μ - 3σ and μ + 3σ.

Hence:

μ - 3σ

100 - 3(13)

= 100 - 39

= 61 mph

μ + 3σ

= 100 + 3(13)

= 100 + 39

= 139 mph

Therefore, the interval is between 61 mph to 139 mph. Option b is correct