c The new mean can be found by adding the increased values to each data score.
A
a See solution.
B
bMore Consistent: Team B Explanation: The range and the standard deviation of Team B are less than those of Team A.
C
cWinner: Team B Explanation: See solution.
Practice makes perfect
a Consider the scores of a bowling match for the following teams.
Team A: 172, 130, 173, 212
Team B: 136, 184, 168, 192
We are told that the team with the greater mean score wins. The mean of a numerical data set is the sum of the data divided by the number of data values. To find out which team wins, let's first calculate the mean for Team A and then for Team B.
Since Team A's mean is greater, Team A wins. Now we will find which team wins if the median score is taken instead. The median of a numerical data set is the middle number when the values are written in numerical order. If the data set has an even number of values, the mean of the two middle values will be the median. Let's first order the data.
Team A: 130, 172, 173, 212
Team B: 136, 168, 184, 192
For both data sets we have an even number of values. Therefore, the median of each data set will be the mean of their middle values.
Team A's Median:&172+ 173/2=172.5
Team B's Median:&168+ 184/2=176
This time Team B wins, since their median score is greater. Therefore, if the rule were changed to say the team with the greater median score wins, the result would be different.
b In this part we need to compare the range and the standard deviation of the data sets to state which team is more consistent. Let's first calculate these measures for Team A.
Team A
The range is the difference of the greatest value and the least value. In this case, the greatest value is 212 and the least value is 130.
Range: 212-130=82pins
Now, let's find the standard deviation. The standard deviation σ of a numerical data set is given by the following formula.
In this formula n is the number of data values in the data set, x_1, x_2,..., x_n are the data values, and x_1-x, x_2-x,... x_n-x are the deviations of each data value. The deviation is given by the difference of the data value and the mean of the data set. We already found out the mean of the data set.
x=687/4 ⇒ x=171.75
With this value we can now calculate the deviation and the square
of the each deviation. Let's do this in a table.
x
x
x-x
(x-x)^2
172
171.75
0.25
0.625
130
171.75
-41.75
1743.0625
173
171.75
1.25
1.5625
212
171.75
40.25
1620.0625
Let's find the mean of the squared deviations. This is called the variance. Let's do it!
Finally, we will take the square root of the variance to get the standard deviation.
σ=sqrt(841.1875) ⇒ σ≈ 29.00
Therefore, the standard deviation is about 29 pins. This means that Team A's scores differ from the mean by about 29 pins.
Team B
Again, we will calculate the range and the standard deviation for Team B. This time the greatest value is 192 and the least value is 136.
Range: 192-136=56pins
The mean for Team B is x=170. Let's use this value to calculate the deviation and the square of each deviation.
Finally, we will take the square root of the variance to get the standard deviation.
σ=sqrt(460) ⇒ σ≈ 21.45
Therefore, the standard deviation is about 21.45 pins. This means that Team B's scores differ from the mean by about 21.45 pins.
Comparison
Now that we have found the range and the standard deviation for each team, we can compare them to find which one is more consistent.
Measure
Team A
Team B
Range
82
56
Standard Deviation
29.00
21.45
Looking at the values, we can see that both the range and the standard deviation of Team B are less than those of Team A. This means that Team B's scores are less spread out. Therefore, Team B is more consistent.
c In another match Team A's score increases by 15 and Team B's scores increases by 12.5 %. Recal that when a real number k is added to each value in a data set, the measures of center can be found by adding k to the original measures of center. With this in mind, we can calculate the new mean for each team to find which one wins.
Team A's Mean:x&=171.75+15
Team B's Mean:x&=170+0.125(170)
&⇓
Team A's Mean:x&=186.75
Team B's Mean:x&=191.25
In this match Team B's mean is greater. Therefore, Team B wins.
