Use Chebyshev s Inequality to Describe Any Set of Data
-At a tennis tournament a statistician keeps track of every serve. The statistician reported that the mean serve speed of a particular player was 99 miles per hour (mph) and the standard deviation of the serve speeds was 15 mph. If nothing is known about the shape of the distribution, give an interval that will contain the speeds of at least eight-ninths of the player s serves.
A) $54 \mathrm { mph }$ to $144 \mathrm { mph }$
B) $69 \mathrm { mph }$ to $129 \mathrm { mph }$
C) $39 \mathrm { mph }$ to $159 \mathrm { mph }$
D) $144 \mathrm { mph }$ to $189 \mathrm { mph }$

Question

Use Chebyshev s Inequality to Describe Any Set of Data
-At a tennis tournament a statistician keeps track of every serve. The statistician reported that the mean serve speed of a particular player was 99 miles per hour (mph) and the standard deviation of the serve speeds was 15 mph. If nothing is known about the shape of the distribution, give an interval that will contain the speeds of at least eight-ninths of the player s serves. 
A) $54 \mathrm { mph }$ to $144 \mathrm { mph }$
B) $69 \mathrm { mph }$ to $129 \mathrm { mph }$
C) $39 \mathrm { mph }$ to $159 \mathrm { mph }$
D) $144 \mathrm { mph }$ to $189 \mathrm { mph }$

Quizplus · Accepted Answer

Chebyshev's Inequality states that for any real number $k > 1$, at least $1 - \frac{1}{k^2}$ of the data values lie within $k$ standard deviations of the mean. To contain at least eight-ninths ($\frac{8}{9}$) of the serves, we set $1 - \frac{1}{k^2} = \frac{8}{9}$, solving for $k$ gives $k = 3$. Thus, the interval is mean ± 3 standard deviations: $99 \pm 3(15) = 99 \pm 45$, which is $54 \mathrm{mph}$ to $144 \mathrm{mph}$.

Use Chebyshev S Inequality to Describe Any Set of Data 54mph54 \mathrm { mph }54mph

Use Chebyshev S Inequality to Describe Any Set of Data $54 \mathrm { mph }$