URBS 362 Test 1

Definition of population: … a collection of units (people, places, objects, procedures, etc.) that are being studied.
Different concept than in urban planning

Definition of unit: a member of a population

Definition of observation: fact or figure we collect about a given unit. It can be expressed as a number or as a quality

Internal source of data
Record or files of an institution undertaking a study
Ex: Statistics Canada
Origin of data is well known
Sources of errors are well known

External source of data
Data obtained from an organization external to the institution undertaking a study
Origin not always known (sampling, calibration, training of experimenters)
Variable adequately defined? (e.g., shoe size, air pollution)
Manipulation of data? (raw data? grouping?)
Ex: data on montreal over time

Interval
Values along a scale with equal distance
Proportionality does not exist (4°C is not twice 2°C) → Calculation of ratio is not relevant
Examples: level of satisfaction (Likert scale), temperature (°C, °F)

Ratio
Values having a true zero
Proportionality exists (4 grams is twice of 2 grams)
Examples: concentration of pollutants, attendance to a consultation

Nominal
Data that represents count in different categories
Examples: Colour, race, religion

Ordinal
Data that represents count in categories with a logical order (but the distance does not matter)
Examples: Age classes, population density classes

1st: calculate z score
2nd: find the z-score of the observation in a normal table
3rd: determine the probability using the probability corresponding to the z-score

if the population standard deviation is known, use the z-distribution. If the population standard deviation is not known, use the t-distribution.

t distribution is used with samples

1) Determine the probability you are interested in
2) Find the number of observations of your sample
3) Calculate the number of degrees of freedom (df) ➢ df = number of observations – 1
4) Find the t-score in the table

Notation
Population: Greek letters
Sample: Roman letters

Calculation of measures of dispersion varies according to the origin of the observations

The theory, methods, and practice of forming judgements about the parameters of a population and the reliability of statistical relationships, typically on the basis of random sampling.
The theory, methods: statistical tests based on a distribution (e.g., Normal, Student’s t)

Definition: … the use of sample data to calculate an interval of possible (or probable) values of an unknown population parameter …

Use: provide two bounds (a lower and an upper bound) of a sample descriptor (e.g., mean) with a certain prespecified confidence level

1. Specify the confidence level (sometimes designated as α)
2. Find the t-score associated with the confidence level
3. Calculate the confidence interval of the mean

Need to find the probability of the upper-bound
100%-2.5% = 97.5% or 0.975 (here, a confidence level of 95%)
Need to find the t-score to calculate the bounds
Ex: for 90% confidence
100%-90% = 10%
10%/2 = 5%
100%-5%=95% (or 0.95) step required to adapt to our table)

then use the table to get t-score

1. Formulate statistical hypotheses (from working hypothesis or from the objective of the study)
2. Determine a significance level (𝛼)
3. Determine the critical value
4. Calculate the test statistic (e.g., t-score)
5. Compare the test statistic with the critical value
6. Accept or reject the null hypothesis (H0 )

H0
Always reflects the absence of difference or the absence of an impact
Also called “null hypothesis”
Example: Mean asphalt thickness is equal to 65 mm, H0: μ = 65 mm
Ha
Always reflects a difference or an effect
Also called “alternative hypothesis” (or H1 )
Example 1: Mean asphalt thickness is different from 65 mm, Ha : μ ≠ 65 mm

The value from a table (e.g., Student’s t table) allows the use in different contexts (urban science, biology, …)

The critical value depends on:
a. The alternative hypothesis (Ha )
Being smaller or larger → one tailed test
Mean being different → two-tailed test
b. The significance level
One tailed: probability as 1- 𝛼
Two tailed: probability as 1 - 𝛼/2
c. The number of degrees of freedom (for several distributions)
Use df and probability to get t score

Accept null: positive and lower than or equal to critical, or negative and greater than or equal to the critical value
Reject null/ accept alternative: positive and greater than the critical value or negative and lower than the critical value

1. Calculation of the pooled s2
2. Calculation of the test statistic