Two Proportions test
A collection of data between peoples’ geographical origin and the relevance of wearing winter
shoes was done to see if there is a connection between peoples’ usage of winter shoes and
where they are from. In this case people were asked about where they are from and were given
three options, the third option being “Other” which was neglected and removed as this eases
the statistical measurement of this question. The two options people were left with were
“Norrland” and “Svealand/Götaland”. These are the three major counties of Sweden and it
makes the most sense to use these as a measurement of geography as a two-option question
due to geographical area.
Mentioned above, the option to answer “Other” was neglected which means that our sample
size shrunk from 51 completed surveys to 44. A two-proportions test was then performed to
spot a possible relevance or difference between origin and usage of proper winter shoes. In our
case the question led “If you are not wearing winter shoes, have you done something with your
day to day shoes, so they can handle the winter better? For instance, ice bugs, extra soles in the
shoes etc. The results showed that 64% of the people from Norrland and 63% of the people
from Svealand/Götaland answered that they wear winter shoes
Conclusion
So, to see if there was a connection between where people are from and their usage of winter
shoes, a Chi-square test and a Two Proportions test was used. Errors that have or may have
affected the results are errors that occurred with the sampling method. Also, systematic errors
such as the questionnaire's design with poorly formulated questions and a lack of definition
towards what is defined as winter shoes and not. The sample had partial non-response errors
and possible processing errors, which are problems that may occur during the preparation of
the data, these are likely errors that have had some effect on the result, either as type I error or
type II error.
The conclusion of the chi-square test is that there is no association between where a person is
from and if they use winter shoes. The result of the chi-square test shows that there is no
empirical evidence, simply stating, that there is no relationship between where a person is
from and if they use winter shoes with a significance level of 5 %. The conclusion is based on
the P-value, (0,738 > 0,05). This could be a type II error if Ha would be true, although to
disprove this sample test it would need a larger sample of people. Hence there is no evidence
to reject H0. One of the problems in the chi-square test was that the cell contents were lower
than five, which meant that the test did not work. The two proportions test proves that there is
little percentile difference between the people wearing winter shoes and where they are from.
The calculated percentages are 63% as well as 64% which entails that geographic origin has
little relevance to the test itself.
The results displayed in both tests that there is no relationship between where a person is
from and their usage of winter shoes. A possibility is that with a larger sample and with a
definition of what winter shoes are, the result could be different.
1
Appendix
Workflow for statistical hypothesis testing
Workflow for test 1 Chi- square test
1. H0 there are no association between variable, Ha, there is an association between variables.
Variables that have been used in chi-square test is where a person is from and if they are
using winter shoes.
2. If H0 is true there is no relationship between were a person is from and if they use winter
shoes, this will be confirmed with a high P-value. The alternative hypothesis, Ha, is that there
is an association between categorical variables, that means that there is an association
between where a person is from and their habits concerning usage of winter shoes.
3. Significant level is 5%, a=0,05 df (2) = 5.99
degree of freedom is calculated by x2=((r-1) (c-1))
4. Decision rule:
P-value method, which mean that P-values < 0.05, H0 will be rejected. The rejection
region, if observed value (calculated by x2= (observed- expected)2 /expected) >5.99 it
will reject H0.
5. Observed p-value is 0,738 and observed value, 0,607
p-value < 5%, H0 is rejected
Observed value > then 5.99 H0 is rejected.
6.
There is no empirical evidence, on 5% significance level that there is a relationship
between were a person is from and if they use winter shoes.
No
Yes
Norrland
6
4,902
19
20,098
25
Other
1
1,373
6
5,627
7
Svealand/Götaland
3
3,725
16
15,275
19
10
41
51
All
All
Pearson Chi-Square = 0,607; DF = 2; P-Value = 0,738
Likelihood Ratio Chi-Square = 0,612; DF = 2; P-Value = 0,736
2
Workflow for test 2 Two proportions test
Null hypothesis
H₀: p₁ - p₂ = 0
Alternative hypothesis
Ha: p₁ - p₂ ≠0
Difference: p₁ - p₂
Null hypothesis
H₀: p₁ - p₂ = 0
Alternative hypothesis
H₁: p₁ - p₂ ≠ 0
Method
Z-Value
P-Value
Normal approximation
0,06
0,954
Fisher's exact
1,000
16 out of 25 people from Norrland answered that they use winter shoes, which is 64%, and
12 out of 19 people from Svealand/Götaland answered that they use winter shoes, which is
63%. Statistically, this proves that there is very little percentile difference between the
variables when calculated since the difference is 1%. From this we can draw the conclusion
that there is no association between geographic origin and the usage of winter shoes.
1. H0: P1-P2=0 H0 is that there aren't any significant differences between the two populations,
Norrland and Svealand/Götaland.
Ha: P1-P2  0 That there is a difference between these two proportions.
2. If H0 is true there is no difference between were a person is from and if they use winter
shoes, this will be confirmed with a high P-value. Ha is when there is an relationship between
a person is from and there usage of winter shoes.
3. The significance level for the test is α=0,05 which is 5%. The observed z-value is 0,06
4. a) The P-value method is used when determining the rejection of H0. For instance, if the
p-value<significance level, then reject H0. In this case, the p-value is 0,954>0,05, therefore
it cannot be rejected.
4. b) The Z-value 0,06 is equal to 0,52392 in the standard normal table. Then subtract
0,52392 from 1 which makes the critical region 0,47608. Every value exceeding 0,47608
will then be rejected.
5. Observed p-value is 0,954 and observed value, 0,06
p-value > 5%, H0 is not rejected
Observed value > 0,47608 then H0 is rejected.
6. There is no empirical evidence, on 5% significance level that there is a relationship
between where a person is from and if they use winter shoes.
3
Cover letter
Umeå university
Please take some time (approximately 2 minutes) to complete this questionnaire.
The purpose of this survey is to find out if students at Umeå University use winter shoes
during the winter and if not, why?
This survey is voluntary and it is confidential. The answers will provide results for the
number of students that wear winter shoes during the winter and the reason behind it, and if
there are some differences from where in Sweden they are from.
The questionnaire
Your gender
- Male
- Female
- Prefer not to say
Your age: Specific number
Where are you from? If you don’t know which county, please write the closest city or if not
from Sweden write country.
- Norrland (Gästrikland, Hälsingland, Härjedalen, Jämtland, Medelpad, Ångermanland,
Västerbotten, Norrbotten and Lappland.)
- Svealand (Dalarna, Närke, Södermanland, Uppland, Värmland and Västmanland.)
- Götaland (Blekinge, Bohuslän, Dalsland, Halland, Skåne, Småland, Västergötland,
Östergötland, Gotland and Öland.)
- Other (Type)
Did you own a pair of winter shoes before you started studying at Umeå University?
- Yes
- No
Have you bought a pair of winter shoes after you moved to Umeå?
- Yes
- No
If you own a pair of winter shoes, have you used them? (Non-mandatory question)
- Yes
- No, none of the mentioned reasons below
- No, too hot during the day
- No, due to fashion purposes
- No, due to comfort issues
4
Are you planning to buy a pair?
- Yes
- No
If you are not wearing winter shoes, have you done something with your day to day shoes so
they can handle the winter better? For instance, ice bugs, extra soles in the shoes and etc.
- Yes
- No
- I’m using winter shoes
Referens
Umeå University (o.d) Umeå University in Figures, Quick Facts 2016, Collected 20 February
2018, from Umeå University, http://www.umu.se/english/about-umu/facts/figures
Comments of the Opposition
Our purpose was to see if student at Umeå University was using winter shoes and under Method
and possible errors there was a wrongful formulation what we meant to write was "thousands
of students" instead of “1000 students”, which we will now change it to. We will also add a
reference to strengthen our claim regarding the amount of students at Umeå University.
In the purpose, it has now been clarified that the report will see if there is an association between
where a person is from and their usage of winter shoes.
In the hypothesis test for two proportions, under 4b, it has now been clarified which order the
numbers have been subtracted in.
5