Q. 3.212 - Elementary Statistics

Question

Dart Doubles. The top two players in the $2001-2002$ Professional Darts Corporation World Championship were Phil Taylor and Peter Manley Taylor and Manley dominated the competition with a record number of doubles. A double is a throw that lands in either the outer ring of the dartboard or the outer ring of the bulls-eye. The following table provides the number of doubles thrown by each of the two players during the five rounds of competition, as found in Chance (Vol. $18$. No. $3$. pp. $48-55$)

a. Obtain the individual population means of the number of doubles. 

b. Without doing any calculations, decide for which player the standard deviation of the number of doubles is smaller. Explain your answer.

c. Obtain the individual population standard deviations of the number of doubles.

d. Are your answers to parts (b) and (c) consistent. Why or why not?

Step-by-Step Solution

Answer

(a) Mean for Taylor and Manley is, $\mu =17.8$

(b) The Taylor has satisfied the given conditions.

(c) Standard deviation of Taylor is, $\sigma =2.64$ and that of Manley is, $\sigma =7.6$

(d) Yes answer to part (c) and part (d) is consistent.

Part (a) Step 1: Given information

We are given that,

Taylor,

The no. values of the data set are, $N=5$

And the data set is,

 $x_i=21, 18, 18, 19, 13$

Manley,

The no. values of the data set are, $N=5$

And the data set is,

 $x_i=5, 24, 20, 26, 14$

Part (a) Step 2: Simplify

We know that, 

Mean is the sum of all values divided by a number of values.

Taylor,

The mean is, 

   $\mu =\frac{\sum _{}{x}_i}{N}=\frac{21+18+18+19+13}{5}=17.8$

Manley,

The mean is,

 $\mu =\frac{\sum _{}{x}_i}{N}=\frac{5+24+20+26+14}{5}=17.8$

Part (b) Step 1: Given information

We need to find out the individual whose standard deviation is the number of doubles is smaller.  

Part (b) Step 2: Explanation

We expect Taylor to have a smaller standard deviation than Manley because its range is much smaller than the range of values for Manley.

Part (c) Step 1: Given information

We need to find out the standard deviation for both the cities. 

Part (c) Step 2: Simplify

As we know that standard deviation is the square root of sum of squared deviation divided by no. of values. 

Taylor,

The population standard deviation is, 

$$\begin{gathered} \sigma =\sqrt{\frac{\sum _{}^{}{\left(x_i-\mu \right)}^2}{N}} \\ =\sqrt{\frac{{\left(21-17.8\right)}^2+{\left(18-17.8\right)}^2+{\left(18-17.8\right)}^2+{\left(19-17.8\right)}^2+{\left(13-17.8\right)}^2}{5}} \\ =2.64 \end{gathered}$$

Manley,

The population standard deviation is, 

  $$\begin{gathered} \sigma =\sqrt{\frac{\sum _{}^{}{\left(x_i-\mu \right)}^2}{N}} \\ =\sqrt{\frac{{\left(5-17.8\right)}^2+{\left(24-17.8\right)}^2+{\left(20-17.8\right)}^2+{\left(26-17.8\right)}^2+{\left(14-17.8\right)}^2}{5}} \\ =7.6 \end{gathered}$$

Part(d) Step 1: Given information

We need to find out whether parts (b) and (c) are consistent or not. 

Part (d) Step 2: Explanation

As we can interpret from parts (b) and (c). part (b) agrees with (c) because standard deviation measures the deviations of the mean and thus standard deviation is dependent upon range.