MATH 1280 wa2
docx
keyboard_arrow_up
School
McGill University *
*We aren’t endorsed by this school
Course
1280
Subject
Mathematics
Date
Jan 9, 2024
Type
docx
Pages
4
Uploaded by antifan
MATH 1280-01
Written Assignment Unit 2
Hitesh Verma (Instructor)
Forty randomly selected students were asked the number of pairs of sneakers they owned.
Let X = the number of pairs of sneakers owned. The results are as follows:
Pairs of sneakers owned
X Frequency
1
2
2
5
3
8
4
12
5
12
6
0
7
1
1.
Find the mean
2.
Find the samples standard deviation,
s
.
3.
Complete the Relative Frequency column and the Cumulative Relative Frequency
Column.
4.
Find the first quartile.
5.
Find the median.
6.
Find the third quartile.
7.
What percent of the students owned at least five pairs?
8.
Find the 40th percentile.
9.
Find the 90th percentile.
Mean
The mean, denoted by μ, represents the average value of all observations in the dataset. To
calculate the mean, we sum up all the values of X and divide by the total number of observations:
Mean (μ) = (ΣX)/n = (122 + 53 + 8 + 4 + 1 + 2 + 5 + 1 + 2 + 6 + 0 + 7 + 1)/40 = 3.78
Sample Standard Deviation
The sample standard deviation, denoted by s, measures the variability or dispersion of the data
around the mean. To calculate the sample standard deviation, we first calculate the squared
deviations from the mean for each observation:
(X - μ)^2 = (122 - 3.78)^2 + (53 - 3.78)^2 + ... + (7 - 3.78)^2
Then, we sum up these squared deviations and divide by n - 1:
Variance (σ^2) = Σ(X - μ)^2 / (n - 1)
Sample Standard Deviation (s) = √(Variance)
Sample Standard Deviation (s) = √(10,624.72 / 39) = 1.29
Relative Frequency and Cumulative Relative Frequency
Relative frequency is the proportion of observations that fall within a given category. Cumulative
relative frequency is the proportion of observations that fall within a given category or any
preceding categories.
Pairs of sneakers
owned
Frequency
Relative
Frequency
Cumulative Relative
Frequency
1
2
0.050
0.050
2
5
0.125
0.175
3
8
0.200
0.375
4
12
0.300
0.675
5
12
0.300
0.975
6
0
0.000
0.975
7
1
0.025
1.000
Quartiles
Quartiles divide the data into four equal groups. The first quartile (Q1) represents the value
below which 25% of the observations fall. The median (Q2) represents the middle value of the
data when the observations are arranged in ascending order. The third quartile (Q3) represents
the value below which 75% of the observations fall.
To find the quartiles, we first order the data in ascending order:
1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 7, 8, 12, 53
Since we have 40 observations, the median will be the average of the 20th and 21st observations:
Median (Q2) = (5 + 5)/2 = 5
To find the first quartile (Q1), we identify the value below which 25% of the observations fall.
Since we have 40 observations, the first quartile will be the 10th observation:
First Quartile (Q1) = 3
To find the third quartile (Q3), we identify the value below which 75% of the observations fall.
Since we have 40 observations, the third quartile will be the 30th observation:
Third Quartile (Q3) = 5
Percent of Students with at Least Five Pairs
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
To determine the percentage of students who owned at least five pairs of sneakers, we sum up
the frequencies for all categories with five or more pairs and divide by the total number of
students:
Percent of students with at least five pairs = (12 + 12 + 0 + 1)/40 * 100% = 31.25%
40th Percentile
The 40th percentile represents the value below which 40% of the observations fall. Since we
have 40 observations, the 40th percentile will be the 16th observation:
40th Percentile = 4
90th Percentile
The 90th percentile represents the value below which 90% of the observations fall. Since we
have 40 observations, the 90th percentile will be the 36th observation:
90th Percentile = 6
This means that 90% of the students owned six pairs of sneakers or less.
Reference:
Problem No 114. Illowsky, B., Dean, S., Birmajer, D., Blount, B., Boyd, S., Einsohn, M.,
Helmreich, Kenyon, L., Lee, S., & Taub, J. (2022). Introductory
statistics. openstax.
https://openstax.org/books/introductory-statistics/pages/2-bringing-it-
together-homework
.