Statistics Flashcards
(196 cards)
1
Q
All variables can be categorised into two
A
Numerical data
Categorical data
2
Q
Numerical data can be broken down into two
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Continuous
Discrete
3
Q
Categorical data can be broken down into two
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Regular categorical
Ordinal
4
Q
Numerical
A
Numbers
5
Q
Numerical
A
Numbers
6
Q
What are two types of variables
A
Numerical
Categorical
7
Q
What are the two types of numerical variables
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Continuous
Discrete
8
Q
What are the two types of categorical data
A
Regular categorical
Ordinal
9
Q
Numerical
A
Numbers
10
Q
Categorical data
A
Data that is sorted into categories such as fav colour
11
Q
Continuous data ( numerical)
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Data that can take any specific value eg: 24cm
12
Q
Discrete
A
13
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Discrete data ( numerical)
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Data that can increase the number of value eg: class interval between 12-30
14
Q
Regular categorical data
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Types of data divided into groups such as: race, age
15
Q
Ordinal data ( categorical)
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Non numerical piece of information with implied order eg: medium, hot, mild
16
Q
Explanatory variable ( independent variable)
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Used to predict changes in another variable (dependant variable).
Like cause n effect
17
Q
Observational studies
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Where researchers observe subjects without manipulating treatments or assigning treatments
18
Q
Experiment
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Is a study where researchers deliberating manipulate one or more variables to observe the response on a response variable
19
Q
Variables that aren’t associated are called
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Independent variables
20
Q
Variables that show some sort of connection with one another
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Dependant variables
21
Q
Association does cause
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Causation
22
Q
Census
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Sample of the population
23
Q
Problems with census
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- people may find it hard to measure as their may be outliers
- population changes, hard to have accurate census
- taking census complicated than sampling
24
Q
Exploratory analysis
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Process of analysing data to summarise main features, using visual methods before applying formal models or testing hypothesis
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Inference
Process of Drawing conclusions about a population based on data from a sample
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Anecdotal evidence
Informal evidence based on personal stories, individual cases rather than scientific data or systematic research
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The sample needs to representative of the entire population
For inference to be valid
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Non response
Occurs when individuals selected for a survey or study doesn’t respond, leading to missing data and potential bias in results
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Voluntary response
Occurs when sample consists of people who volunteer to respond because they have strong opinions on the issue
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Convience sample
Non- random sample made up of individuals who are easy to reach out to. That can lead to bias results
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What are the three random sampling techniques?
- simple
- stratified
- cluster sampling
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Stratified sampling
The population is divided into groups (strata), and a random sample is taken from each group.
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Simple sampling
Every individual in the population has an equal chance of being selected
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Cluster sampling
The population is divided into groups (clusters), some clusters are randomly selected, and all individuals within chosen clusters are surveyed.
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Placebo
Fake treatment used as the control group for medical studies
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Placebo effect
Participants behaviour caused by belief and receiving the treatment not the treatment itself
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Blinding
When experimental units don’t know whether they are in the control or treatment group
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Double blind
When both experimental units and the researchers who interact with patients don’t know who the control group of who is in the treatment group
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Steps in experiment
1. Control for the potential effect of other variables
2. Randomise : randomly assigning subjects to treatments and random sample from population
3. Replicate : replicate by collecting a large sample/ entire
4. Block: if variables affect the response
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Scatterplots
Type of graph that displays data on a two dimensional plane shows relationship between two variables
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What is the correlation between children per woman and fertility rate?
Negative correlation
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What is correlation?
A statistical measure that describes the strength and direction of the relationship between two variables.
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What is a positive correlation?
As one variable increases, the other also increases.
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What is a negative correlation?
As one variable increases, the other decreases.
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What is no correlation?
No predictable relationship between the variables.
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What does a correlation of 1 mean?
Perfect positive correlation.
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What does a correlation of -1 mean?
Perfect negative correlation
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What does a correlation of 0 mean?
No correlation
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Dot plot
Graph that displays individual data points as dots along a number line
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Darker areas represent where there are more
Observations
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Sample mean formula and def
Average set of values from a sample of a population
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Population mean formula
Average of all values in an entire population
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Why is the sample mean sample static?
Because it’s calculated from a sample
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Why does sample mean serve as a point estimate of population mean?
Provides estimate of population true average
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Stacked dot plot
displays data points as dots stacked vertically for identical values, making it easy to see the frequency of each value in the dataset
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Histogram
is a bar chart that displays the frequency of data within specified intervals (bins), used to visualize the distribution of continuous data
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There is three shapes of distributions
Unimodal
Bimodal/ multimodal
Uniform
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Unimodal distribution
Is data with a single peak or mode
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Biomodal distribution
Is data with two peaks
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Multimodal distribution
Is data with multiple peaks
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Uniform distribution
Evenly spread data where all outcomes are likely
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The two types of skewness are
Right skewed
Left skewed
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Right skewness ( positive skew)
Majority of data is on the left, right tail (larger values) is longer
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Left skewness ( negative skew)
Left tail ( smaller values) is longer and majority of data is on the right
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Sample variance
Spread of data points in a sample
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We square deviations to
Get rid of negatives so observations equally distant from mean are weighed equally
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Absolute value
Leaves the sign of the number and leaves its only magnitude making it non negative
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Sample standard deviation
The average distance of data points from sample mean. Square root of variance
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Median
Is the middle value if the data set when ordered in ascending
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Median is midpoint of data
Known as 50th percentile
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Quartile 1
25th percentile
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Quartile 3
75th percentile
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Interquartile range (IR)
Difference between third quartile and first quartile. Identifies central portion where data lies
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Box plot
Graphically definition of data set shows median, quartiles and outliers
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Outliers
Data points that are significantly different from rest of data in the data set
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Whiskers
Show range of data
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Benefits of outliers
- helps indentify extreme skew in distribution
- indemnifies data collection and entry errors
- provides insight into interesting features of data
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Random processes
Sequence of outcomes that evolve over time with uncertainty where each result is determined by chance
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Frequentist probability
Proportion of times an event occurs in repeated experiments
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Bayesian interpretation
Probability as a measure of belief based on prior knowledge and updated with new evidence
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Law of large numbers R
States that as sample increases the sample mean will get closer to the population mean
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The Gamber’s Fallacy
Mistaken belief that past random events affect the probabilities of future random events
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Mutually exclusive (disjoint) outcomes
Are outcomes that cannot happen at the same time.
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Example of a mutually exclusive outcome
Flipping heads and tails at the same time
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Non- disjoint outcomes
Outcomes that can happen at the same time. Might influence each others likelihood
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Example of a non disjoint outcome
Drawing a card that is queen and red from
The deck
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Rules of probability distributions
1. Events listed must be disjoint
2. Each probability must be between 0 and 1
3. Probability must be total one
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Complementary events
are two events where one event represents all outcomes that are not in the other event.
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Sample space
The complete set of all possible outcomes in a probability experiment.
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Two outcomes can be DEPENDANT
When the outcome of one affects the probability of the other
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Two ouctomes can be INDEPENDENT
outcome of one doesn’t change the probability of other
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What does a difference in conditional probabilities suggest?
It may indicate dependence between two variables.
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What is the next step after noticing a difference in conditional probabilities?
Conduct a hypothesis test to check if the difference is due to chance
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What does a large difference in conditional probabilities indicate?
Stronger evidence that the variables are dependent
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Why is sample size important in determining dependence?
A large sample size means even small differences can provide strong evidence of dependence
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P(B|A)
Probability of B given A has already happened
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Conditional probability
Probability of an event occurring given that another event has already happened. Eg: P(B|A)
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Product rule used
When probability of both events happening
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Addition rule used
Probability of either event happening
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Normal distribution
Unimodal and symmetric and bell shaped
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Percentile
Percentage of observations that are all below a given data point
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Z-score
How many standard deviations an observation is from the mean
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Z score can be used for any distributions but
Only in a normal distribution can be used to calculate percentiles
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Observations |z| > 2
Are typically considered unusual
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Z score formula shows
How many standard deviations x is from the mean
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Empirical Rule
68% of data lies within 1 standard deviation of the mean
95% of data lies within 2 standard deviations
99.7% lies within 3 standard deviations
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Percentile is the area below
The probability distribution curve to the left of the graph
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Six sigma
Processes that stay within 6 standard deviations from the mean, ensuring high quality
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Geometric distribution
Gives the probability that the first success happens on trial K
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Geometric probability
Gives the chance that the first success happens on the k-th trial
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Binomial Distribution
describes the probability of getting exactly k successes in n independent Bernoulli trials, where each trial has the same probability p of success
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Bernoulli Trial
An experiment with two outcomes
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Central tendency
Describes the center or typical value in a dataset (e.g., mean, median, mode)
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Variation
Measures how spread out the values are (e.g., range, variance, standard deviation, IQR)
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If distribution is symmetric
The center is defined as the mean= median
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If data is extremely skewed
Can transform them into modelling such as the log transformation
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Skewed data after transformed will be easier to model
As the outliers become less prominent
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Intensity Map
A map where darker colors represent higher values or frequencies of data
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Contingency Table
Type of table used to display the frequency distribution of two categorical variables. Shown by organises the variables into rows and columns
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Bar plot
displays categorical data with bars representing the frequency or value of each category
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A relative frequency bar plot
shows proportions (percentages) instead of raw counts for each category. It helps compare categories on a relative scale.
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The difference between bar plots and histograms
Bat plots are categorical and numerical
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Stacked bar plot
Bars stacked to show counts from different groups
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Side-by-side bar plot
Bars placed next to each other for comparison
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Standardized bar plot
Bars show proportions (percentages), not counts
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Null hypothesis (H₀)
Statement that assumes no effect or no difference
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Alternative hypothesis (H₁ or Ha)
Statement that claims there is an effect or a difference
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Point estimate
A single value used to estimate an unknown population parameter
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Example of point estimate
If you take a sample and calculate the sample mean, that value is a point estimate of the population mean.
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Margin of error
A range within which true population parameter is expected to lie example : 41% +_ 2.9% so true proportion is between 38.1% and 43.9%
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Confidence level
Probability that true population parameter lies within the margin of terror eg: 95% confidence
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error in estimate
refers to the difference between the estimated value (from a sample) and the actual true value (of the population)
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Bias
systematic error that leads to incorrect or skewed results.
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Sampling error
natural variation that occurs when you use a sample instead of the whole population
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Sampling error
natural variation that occurs when you use a sample instead of the whole population. It decreases with larger sample sizes
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Sampling distribution
how a statistic (e.g., sample mean) would vary if you took many random samples from the population
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Central Limit Theorem (CLT)
sampling distribution of the sample mean will be approximately normal (bell-shaped) regardless of the population’s original distribution as long as the sample size is large enough
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The mean of sampling distribution
Equal to population mean
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Standard error
is the standard deviation of the sampling distribution of a statistic ( sample mean)
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CLT conditions
1) independence: sample should be random and the population should be at least n<10% than the sample
2) sample size should be large. For proportions at least 10 successes and 10 failures and for mean n>/ 30
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When u don’t know the true population use the
Sample proportion as best estimate
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If sample size is small it’s likely
The sampling distribution may be skewed. Normal distribution won’t work.
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Confidence Interval (CI)
A range of values used to estimate a population parameter (like a mean or proportion) with a certain level of confidence (e.g., 95%)
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If there is a 95% confidence interval
about 95% of those intervals would contain the true population value
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Hypothesis testing
We examine two competing hypothesis the null and the alternative and based on that data we make a decision on which hypothesis is likely true
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There’s two types of errors
Type 1 error
Type 2 error
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Type 1 error
You reject the null hypothesis when it’s actually true. You assume there is an effect and there isn’t
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Type 2 error
Failure to reject null hypothesis when it’s actually false. You think there isn’t an effect but there is
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There could be errors to rejection
- lack of information
- incorrectly rejecting it
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Type 1 error typically
Set to 0.05
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Conditions for hypothesis testing
1. Respondents should be independent of each other
2. Sampling should be random
3. The Sample size should be less than 10% of the total population
4. There should be atleast 30 respondents
5. There should be atleast 10 expected successes and 10 expected failures
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Test static
Tells u how far your sample proportion is from the null hypothesis proportion and measured in standard errors
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P-value
is the probability of getting results at least as extreme as the observed results, assuming the null hypothesis is true
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Small p-value
means the observed data is unlikely under the null hypothesis, so you reject H0
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Two sided hypothesis test
Checking for population parameter difference in both directions
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One sided hypothesis test
Checking for an increase or decrease in population parameter
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Point estimate
Single value from sample data used to approximation unknown population parameter
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Parameter of interest
Specific population characteristic being estimated/ tested
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Marginal error
Z x SE
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Goodness of a test ( chi-squared test)
To test whether the observed frequencies in categories match the expected frequencies based on a specific theoretical distribution.
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Chi square test has one parameter
That’s degree of freedom
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Degrees of freedom
represent the number of independent values that can vary in a statistical calculation while still satisfying a given constraint.
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Degrees of freedom influences
Shape, centre and spread of distribution
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P value
Tail under the chi square distribution
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Chi square conditions
1. Independence: for each case that contributes a count to the table must be independent for all other cases in the table
2. Sample size: each scenario must have atleast expected cases
3. Df>1 : Degrees of freedom must be greater than 1
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Not checking the condition
Can affect the tests error rates
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One sample
Looking for situations where you’re analysing data from a single group
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Two sample
Comparisons between two separate goods
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Expected counts
are the frequencies you would expect in each category if the null hypothesis were true.
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What was the objective of the Friday the 13th Traffic Study?
To compare traffic flow on Friday the 13th versus the previous Friday (6th) at two UK locations.
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What years did the Friday the 13th Traffic Study cover?
1990 to 1992
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What was the consistent pattern found in the traffic study?
Traffic volume was consistently lower on Friday the 13th compared to Friday the 6th.
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What was the largest difference in traffic volume observed?
4382 fewer cars on March 1992 at location 2
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What statistical distribution was used to analyze the traffic data?
The t distribution, due to small sample size and unknown population standard deviation.
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What is the null hypothesis (H₀) in the traffic study?
Mean traffic volume on the 6th equals mean traffic volume on the 13th
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What key assumption must hold for the t-test in the traffic study?
Independence of traffic flow data between the two dates and locations
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Why use the t distribution instead of normal distribution in small samples?
It accounts for extra uncertainty in the standard error by having thicker tail
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Does the study prove that people believe Friday the 13th is unlucky?
No, it only shows a difference in traffic volume, not beliefs
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What does a confidence interval excluding 0 imply about a hypothesis test?
It supports rejecting the null hypothesis (significant difference).
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What is a correct interpretation of a p-value?
The probability of observing the data (or more extreme) assuming the null hypothesis is true
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What is an incorrect interpretation of a p-value?
That it gives the probability the null or alternative hypothesis is true
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Are reading and writing scores from the same students independent?
No, they are paired since the same students took both tests
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What is the parameter of interest for paired data?
The average difference in scores between reading and writing for all students
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What conclusion if p-value > 0.05 in paired test?
Fail to reject H₀; no convincing evidence of difference in average scores
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What is tested in the diamond price example?
Whether 1-carat diamonds have higher price per point than 0.99-carat diamonds
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What was the conclusion from the diamond price test?
1-carat diamonds have significantly higher prices per point; 0.99-carat diamonds may be better value
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What does linear regression model?
Relationship between one response variable (y) and one explanatory variable (x)
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What is a residual in regression?
Difference between observed and predicted values (residual = observed − predicted)
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What does the correlation coefficient (r) measure?
The strength and direction of the linear relationship between two variables (-1 to +1)
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What is the formula for a regression line?
^
y = b0 + b1 x where b0 is intercept and b1 is slope
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What are key conditions for linear regression?
Linearity, nearly normal residuals, constant variability (homoscedasticity).
193
How is slope interpreted in context?
The expected change in y for a one-unit increase in x
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What does R² represent?
Proportion of variability in y explained by the model (e.g., 0.56 means 56%)
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What is the difference between prediction and extrapolation?
Prediction uses x-values within observed range; extrapolation predicts outside the range (less reliable)
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What is multiple linear regression?
A model predicting y from multiple explanatory variables (x₁, x₂, ...).
