a To calculate relative frequencies, divide each frequency by the grand total, 80.
B
b Use the table from Part A and the table given in the exercise.
C
c Calculate the conditional relative frequencies.
A
aTable:
Play an Instrument
Gender
Yes
No
Total
Female
0.35
0.2125
0.5625
Male
0.25
0.1875
0.4375
Total
0.60
0.40
1
B
bPercent who play an instrument? 60 % , marginal relative frequency. Percent of males who do not play an instrument? 42.9 %, conditional relative frequency.
C
cAssociation between the sex of a student and if they play an instrument? Yes, see solution.
Practice makes perfect
a We need to create a two-way relative frequency table using the given information. To do so, we will divide each frequency by the grand total, 80.
Play an Instrument
Gender
Yes
No
Total
Female
28/80
17/80
45/80
Male
20/80
15/80
35/80
Total
48/80
32/80
80/80
By rewriting the fractions as decimals, we will have a completed relative frequency table.
Play an Instrument
Gender
Yes
No
Total
Female
0.35
0.2125
0.5625
Male
0.25
0.1875
0.4375
Total
0.60
0.40
1
b Now, using the table from Part A, we have been tasked with finding two specific pieces of information.
The percent of the students that play an instrument.
The percent of the males surveyed that do not play an instrument.
Let's think about these two situations one at a time.
What percent of the students play an instrument?
Let's look at the table.
Play an Instrument
Gender
Yes
No
Total
Female
0.35
0.2125
0.5625
Male
0.25
0.1875
0.4375
Total
0.60
0.40
1
We can see that the corresponding relative frequency is 0.6. We can convert this to a percentage by multiplying it by 100.
0.6*100=60
Therefore, 60 % of the students surveyed play an instrument. This is a type of marginal relative frequency because it tells us the portion of the whole that represents one specific characteristic.
What percent of the males surveyed do not play an instrument?
To answer this question, we need to use the original table given in the exercise.
Play an Instrument
Gender
Yes
No
Total
Female
28
17
45
Male
20
15
35
Total
48
32
80
We can see that 15 of the 35 males surveyed do not play an instrument. Therefore,
1535=42.9 %
of the males do not play an instrument. This is a type of conditional relative frequency because it describes a portion of a group with one characteristic that also has another characteristic.
c To see if there is an association between the sex of a student and whether the student plays an instrument, we need to calculate conditional relative frequencies. We have the following.
28 of the 48 students who play an instrument are female. Therefore, 2848= 58.33 % of the students who play an instrument are female.
Meanwhile, 45 of the 80 students surveyed are female. Therefore, 4580= 56.25 % of the students surveyed are female.
Because the percentage of students who play an instrument who are female is higher than the percentage of students who are female,
58.33 % > 56.25 %
we know that female students are more likely to play an instrument.
