1. The Relationship between the Average Amount of Sleep a High school Teenager gets
per School Night and the Total Average amount of Video Games played in one
School Week
Math Studies Internal Assessment
David Batchelor
Due: December 2010
International School Bangkok
Teacher: Mr. DeMille
Word Count:

2. What is the Relationship between the amount of time spent playing video games and
your amount of sleep?
Introduction
In developed countries around the world, many issues arise from playing excessive
video games from addiction to an increase in violent behavior commonly among
younger children. Studies also show that playing too many video games can affect
your sleeping patterns in a negative way. For example, if you play an excessive
amount of video games before bed time, it becomes more difficult to fall asleep.
However, I decided to undertake a similar yet different study to find out how much
video games an average high schooler plays and how this affects how many hours of
sleep he/she gets. There were no online articles for this study so I decided to find out
for myself using the high school population at school. At the beginning of the
experiment I figured that kids who played more video games received less sleep.
Statement of Task
The main intention of this investigation was to find out whether the amount of time
spent playing video games has an effect on the amount of sleep an average high-
schooler gets. To do this, I gathered data at lunch during school hours and asked 50
students how many hours of video games they spent playing on average in an entire
school week and how many hours of sleep they get in an average school night. I asked
a wide range of students from 9th to 12th grade to create an equal dispersion of data.
Throughout the project, I asked a range of students who play a lot of video games,
students who don’t play a lot of video games and students who play a moderate
amount of video games.
Plan of Investigation
I am going to make a scatter plot and make a line of best fit to see if there is a linear
correlation between the data. Afterwards, I will use the Standard Deviation
Calculations to find out the dispersion of the data. With this data, I will find the least
squares regression line to find out the equation of the line. Next, I will find out the
Pearson’s Correlation Coefficient to find the strength of the correlation of my data.
Finally, I will run a Chi Squared test to see if the data is independent or dependent of
one another.

3. Data Collection
Amount of Video Games and Amount of Sleep for 50 Students (hours)
Table 1
Total Average Hours of Video Games in One
School Week
Hours of Sleep on Average School
Night
10 9
2 6.5
0 6
5 5
15 3
8 8
0 8
2 5
2 7
5 7
3 7
3 6
6 8.5
2 8
0 8
0 7
2.5 7
0 8
0 7
0 9
2 7
3 8
0 7
6 7
0 8
0 7
0 8
4 6
6.5 8
5 8
0 7
0 7
1 5
5 6.5
3.5 7
2 7
1 7
0 8.5
0 7.5
7 7
0 7

4. 0 6
0 8.5
1 8
0 8.5
3 9
0 7
3 7
0 8
5 5
Table 1: Table 1 displays the number of hours spent playing video games in one week
and the corresponding amount of sleep. Although the data looks quite scattered in the
table, you can tell that people who played less video games received more sleep.
However this was not the case for some students because they claimed to have a lot of
homework and that resulted in less sleep.

5. Figure 1
Figure 1: Figure 1 shows the relationship between the number of hours spent playing
video games in total in an average school week and the amount of sleep the average
high school teenager received in an average school night. (Graph by Excel)
Standard Deviation
The standard deviation is a formula which measures the dispersion of the data. In this
investigation, I wanted to find the dispersal of the data found between the amount of
time spent playing video games and the amount of sleep.
Formulae:
𝑆𝑥 = √
∑ 𝑥2
𝑛
− 𝑥̅2
and 𝑆 𝑦 = √
∑ 𝑦2
𝑛
− 𝑦̅2
0
1
2
3
4
5
6
7
8
9
10
0 2 4 6 8 10 12 14 16
HoursofSleeponanAverageSchoolNight
Total Average Hours of Video Games in One School Week
Amount of Time Spent Playing Video Games vs.
Sleep

6. 𝑆𝑥 = √
783.75
50
− 6.1009
𝑺 𝒙 = 𝟑. 𝟎𝟗
The standard deviation of x (amount of time spend playing video games in one school
week) is 3.09. This says that the data collected for the amount of time playing video
games is highly dispersed and relatively insufficient data.
𝑆 𝑦 = √
2624.75
50
− 51.1225
𝑺 𝒚 = 𝟏. 𝟏𝟕
The standard deviation of y (amount of sleep in an average school night) is 1.17. This
data is very compact and is sufficient data.
Least Squares Regression
The least squares regression formula calculates the relationship between x and y in the
form of a line equation. This line equation is in y intercept form when completed and
can be fitted on the scatter plot graph as a line of best fit if applicable.
Formulae:
𝑦 − 𝑦̅ =
𝑆 𝑥𝑦
𝑆 𝑥
2
( 𝑥 − 𝑥̅) where 𝑆𝑥𝑦 is the covariance
∑ 𝑥𝑦
𝑛
− 𝑥̅ 𝑦̅
𝑆 𝑥𝑦 =
823.5
50
− (2.47)(7.15)
𝑆 𝑥𝑦 = −1.0905
Hence:
𝑦 − 7.15 =
−1.19
9.574100003
( 𝑥 − 2.47)
𝑦 − 7.15 = −0.1242936673( 𝑥 − 2.47)
𝑦 − 7.15 = −0.1242936673𝑥 + 0.3070053582
𝒚 = −𝟎. 𝟏𝟐𝟒𝒙+ 𝟕. 𝟒𝟔
𝒚 = −𝟎. 𝟏𝟐𝟒𝒙+ 𝟕. 𝟒𝟔 is the least squares regression line. This line can be used as a
line of best fit in figure 1.

7. Pearson’s Correlation Coefficient
This equation is useful when you want to find out how relative the data of your two
variables are.
𝑟 =
𝑆𝑥𝑦
𝑆𝑥 𝑆 𝑦
𝑟 =
−1.19
(3.09)(1.17)
𝑟 = −0.329
𝒓 𝟐
= 𝟎. 𝟏𝟎𝟖

8. Figure 2
This is the same scatter plot graph as figure 1 however this one contains the line of
best fit from the previous calculations.
Figure 2 is the same scatter plot with a line of best fit. As you can see, the equations
above gave me the y intercept formula. The computer generated formula from Excel
gave me the same descending line which shows that the calculations were accurate.
Although the line has a negative linear correlation, it is close to being flat. This shows
that the data collected throughout the experiment was varied and that the two
variables almost had no effect on one another.
0
1
2
3
4
5
6
7
8
9
10
0 2 4 6 8 10 12 14 16
HoursofSleeponanAverageSchoolNight
Total Average Hours of Video Games in One School Week
Amount of Time Spent Playing Video Games
vs. Sleep
𝒚 = −𝟎. 𝟏𝟐𝟒𝒙 + 𝟕. 𝟒𝟔
𝐫 𝟐
= 𝟎. 𝟏𝟎𝟖

9. Chi Squared Test
The Chi-squared test (χ2
) is the test that is used to find out if two variables from the
same sample are independent or dependent of one another. Sample tables below show
the templates used for a Chi-squared test.
Observed Values
A1 A2 Sum
B1 a b w
B2 c d x
Sum y z n
Expected Values
A1 A2 Sum
B1 𝑤𝑦
𝑛
𝑤𝑧
𝑛
w
B2 𝑥𝑦
𝑛
𝑥𝑧
𝑛
x
Sum y z n
χ2
= ∑
(𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 − 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒)
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑉𝑎𝑙𝑢𝑒
Degrees of Freedom measures the amount of data values that can differ
Df = (r - 1)(c - 1)
r= rows c= columns
Null (𝐻 𝑜) Hypothesis: Amount of sleep and hours spent playing video games are
independent (not dependent of one another)
Alternative (𝐻𝐼) Hypothesis: Amount of sleep and hours spent playing video games
are dependent (not independent of one another)
2

10. Table 2: Observed Values
Hours of Sleep
5 – 7.5 7.5 - 9 Sum
0 - 3.5 22 15 37
3.5 – 6.5 5 2 7
6.5 – 10 2 2 4
Sum 29 19 48
Table 2: This table shows the observed values for amount of sleep and hours spent
playing video games put into ranges so that the data is easier to calculate with similar
results
Table 3: Expected Value (calculations from template were carried out here)
Hours of Sleep
5 – 7.5 7.5 - 9 Sum
0 – 3.5 22.0 15.0 37
3.5 – 6.5 4.00 3.00 7
6.5 – 10 3.00 2.00 4
Sum 29 19 48
𝛘 𝟐
= 𝟎. 𝟎𝟏𝟒𝟔
Df = (3 - 1)(2 - 1) => 2 Df
The χ2
critical value at 5% significance with a degrees of freedom of 2 is 5.991. The
chi squared value is 0.0146 which is less than 5.991 (0.0146<5.991). With this in
mind, chi-squared is smaller therefore we do not reject the null hypothesis thus the
amount of sleep and hours spent playing video games are independent.
HoursspentplayingvideogamesHoursspentplayingvideogames

11. Data Understanding
To start off, the graph in figure 1 shows a scatter plot graph with all the data from the
50 students. In figure 2, the same graph was displayed but this time with a line of best
fit. The line of best fit was found with the least squares regression formula. The
equation found within my own calculations was the same as the computer generated
formula from Excel. The line demonstrated a negative linear correlation that was
almost flat. This demonstrates weak trends and almost neutral data.
Furthermore, the standard deviation test showed that the variables were quite
dispersed with 𝑆 𝑥 = 3.09 and 𝑆 𝑦= 1.17.
Limitations
There were a few limitations in my investigation. First, I surveyed 50 students from
an overall population of about 500. Although I attempted to survey students from
every grade level and gender, the data would have been far more accurate if all 500
students were surveyed. I also tried to survey people who I knew played a lot of video
games and those who played very little if not at all and see if there were any relations
between the amount of sleep that each got.
Another limitation is that about half of the students that I surveyed stated that they
hadn’t played video games in weeks due to the homework load. Although some
students didn’t play video games at all, they still got little sleep due to homework
etc… Some of the students who I interviewed played an average of 6 and a half hours
of video games a week and still managed to get 8 hours of sleep a night which is more
that many students get without playing video games.
Conclusion
Although there were many limitations that made my data less accurate than was
expected, I believe that I received a concrete answer from my studies. According to
my theory, as the x axis increases (amount of video games), the y axis should
decrease (amount of sleep). My line of best fit does technically fits this theory
however the strength of the line is weak. If I had asked a completely separate group of
50 students, there could be a great chance that my line went level with no relation or
even up (although highly doubtful). However I believe that if the entire 500 student
population in the high school were surveyed, that the relationship of the data could
have strengthened. In other words, the line of best fit would have turned into a more
negative slope. In conclusion, the more video games the high school population plays,
the less sleep they get on an average school night.