Question

1 (a). The following data set lists a series of ACT scores {8, 12, 15, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 23, 23, 24, 25, 36}. Write down the five number summary and construct a boxplot which includes the position of the upper and lower fences

(b). At a tennis tournament a statistician keeps track of every serve. The statistician reported that the mean serve speed of a particular player was 104 miles per hour (mph) and the standard deviation of the serve speeds was 8 mph. Assume that the statistician also gave us the information that the distribution of the serve speed was bell shaped. What proportion of the player’s serves are expected to be between 112 mph and 120mph?

Answer 1

1 (a). We have to write down the five number summary and construct a boxplot which includes the position of the upper and lower fences of above part (a)

For finding the five number summary we arrange the data in ascending order

8, 12, 15, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 23, 23, 24, 25, 36

N = Total number of observations = 28

The five number summary

1) Smallest value = 8

2) Q1 = first quartile = value of (N + 1)/4 th observation

= value of (28 + 1)/4 th observation  

= value of 7.25th observation

= value of 7th observation + 0.25*[8th observation - 7th  observation] = 17+0.25[18-17] = 17 + 0.25

Q1 = first quartile = 17.25

3) Q2 = Median = second quartile = value of (N + 1)/2 th observation

= value of (28 + 1)/2 th observation =  value of 14.5th observation

= value of 14th observation + 0.5*[15th observation - 14th  observation] = 19 + 0.5*[19-19] = 19

Q2 = Median = second quartile = 19

4) Q3 = third quartile = value of 3*(N + 1)/4 th observation

= value of 3*(28 + 1)/4 th observation  

= value of 21.75th observation

= value of 21th observation + 0.75*[22th observation - 21th  observation]

= 21 + 0.75*[21 - 21] = 21

Q3 = third quartile = 21

5) Largest value = 36

The five number summary is

Five number summary

1) Smallest value = 8

2) Q1 = first quartile = 17.25

3) Q2 = Median = second quartile = 19

4) Q3 = third quartile = 21

5) Largest value = 36

Using this five number summary we construct boxplot using Excel is as below

(b) The values provided in the part (b) are as below

mean = = 104

standard deviation = = 8

We have to find the proportion of the player’s serves are expected to be between 112 mph and 120mph

-------------(1)

We convert the above X into Z using following equation.

----------------------(2)

Using equation (2) in equation (1) we get

We find the above probability using table of Standard Normal Curve Areas

The proportion of the player’s serves are expected to be between 112 mph and 120mph is 0.1359