USING DATA TO ANSWER STATISTICAL QUESTIONS
The information we gather with experiments and surveys is collectively called data. Statistics is the art and science of learning from data. Statistical problem solving consists of four things:

1. Formulate a statistical question
2. Collect data
3. Analyse data
4. Interpret results

The three main components of statistics for answering a statistical question are:

1. Design
   Stating the goal and/or statistical question of interest and planning how to obtain data that will address them. (e.g: how do you conduct an experiment to determine the effects of ‘X’)
2. Description
   Summarizing and analysing the data that are obtained (e.g: summarizing people’s tv-habits in ‘hours of tv watched per day’)
3. Inference
   Making decisions and predictions based on the data for answering the statistical question. (predicting the outcome of an election, based on the description of the data)

Probability is a framework for quantifying how likely various possible outcomes are.

SAMPLE VERSUS POPULATION
The entities that are measured in a study are called the subjects. This usually means people, but it can also be schools, countries or days. The population is the set of all the subjects of interest. In practice, we usually have data for only some of the subjects who belong to that population. These subjects are called a sample.

Descriptive statistics refers to methods for summarizing the collected data. The summaries usually consist of graphs and numbers such as averages and percentages. Inferential statistics are used when data are available from a sample only, but we want to make a decision or prediction about the entire population. Inferential statistics refers to methods of making decisions or predictions about a population, based on data obtained from a sample of that population.

A parameter is a numerical summary of the population. A statistic is a numerical summary of a sample taken from the population. The true parameter values are almost always unknown, thus we use sample statistics to estimate the parameter values.

A sample is random when everyone in the population has the same chance of being included in the sample. Random sampling allows us to make powerful inferences about populations. The margin of error is a measure of the expected variability from one random sample to the next random sample.

The formula for calculating the approximate margin of error is:  . In this case, ‘n’ is the number of subjects.

THE ASSOCIATION BETWEEN TWO CATEGORICAL VARIABLES
When analysing data the first step is to distinguish between the response variable and the explanatory variable. The response variable is the outcome variable on which comparisons are made. If the explanatory variable is categorical, it defines the groups to be compared with respect to values for the response variable. If the explanatory variable is quantitative, it defines the change in different numerical values to be compared with respect to values for the response variable. The explanatory variable should explain the response variable (e.g: survival status is a response variable and smoking status is the explanatory variable).

An association exists between two variables if a particular value for one variable is more likely to occur with certain values of the other variable.

A contingency table is a display for two categorical variables. Conditional proportions are proportions which formation is conditional on ‘x’. A conditional proportion should be conditional to something. A conditional proportion is also a percentage. The proportion of the totals (e.g: percentage of total amount of ‘no’) is called a marginal proportion.

There is probably an association between two variables if there is a clear explanatory/response relationship, that dictates which way we compute the conditional proportions. Conditional proportions are useful in determining if there’s an association. A variable can be independent from another variable.

THE ASSOCIATION BETWEEN TWO QUANTITATIVE VARIABLES
We examine a scatterplot to study association. There is a difference between a positive association and a negative association. If there is a positive association, x goes up as y goes up. If there is a negative association, x goes up as y goes down.

Correlation describes the strength of the linear association. Correlation (r) summarizes th direction of the association between two quantitative variables and the strength of its linear trend. It can take a value between -1 and 1. A positive value for r indicates a positive association and a negative value for r indicates a negative association. The closer r is to 1, the closer the data points fall to a straight line and the stronger the linear association is. The closer r is to 0, the weaker the linear association is.

The properties of the correlation:

• The correlation always falls between -1 and +1.
• A positive correlation indicates a positive association and a negative correlation indicates a negative association.
• The value of the correlation does not depend on the variables’ unit (e.g: euros or dollars)
• Two variables have the same correlation no matter which is treated as the response variable and which is treated at the explanatory variable.

The correlation r can be calculated as following:

N is the number of points.  and ȳ are means and

SUMMARIZING POSSIBLE OUTCOMES AND THEIR PROBABILITIES
All possible outcomes and probabilities are summarized in a probability distribution. There is a normal and a binomial distribution. A random variable is a numerical measurement of the outcome of a random phenomenon. The probability distribution of a discrete random variable assigns a probability to each possible value. Numerical summaries of the population are called parameters and a population distribution is a type of probability distribution, one that applies for selecting a subject at random from a population.

The formula for the mean of a probability distribution for a discrete random variable is:

μ= ΣxP(x)

It is also called a weighted average, because some outcomes are likelier to occur than others, so a regular mean would be insufficient here. The mean of a probability distribution of random variable X is also called the expected value of X. The standard deviation of a probability distribution measures the variability from the mean. It describes how far values of the random variable fall, on the average, from the expected value of the distribution. A continuous variable is measured in a discrete manner, because of rounding. A probability distribution for a continuous random variable is used to approximate the probability distribution for the possible rounded values.

PROBABILITIES FOR BELL-SHAPED DISTRIBUTIONS

The z-score for a value x of a random variable is the number of standard deviations that x falls from the mean. It is calculated as:

The standard normal distribution is the normal distribution with mean  and standard deviation  . It is the distribution of normal z-scores.

PROBABILITIES WHEN EACH OBSERVATION HAS TWO POSSIBLE OUTCOMES
An observation is binary if it has one of two possible outcomes (e.g: accept or decline, yes or no). A random variable X that counts the number of observations of a particular type has a probability distribution called the binomial distribution. There are a few conditions for a binomial distribution:

1. Two possible outcomes
   Each trial has two possible outcomes.
2. Same probability of success
   Each trial has the same probability of success
3. Trials are independent

The formula for the binomial probabilities for any n is:

The binomial distribution is valid if the sample size is less than 10% of the population. There are a couple of formulas for the binomial distribution:

 and

POINT AND INTERVAL ESTIMATES OF POPULATION PARAMETERS
A point estimate is a single number that is our best guess for the parameter (e.g: 25% of all Dutch people are above 1,80m). An interval estimate is an interval of numbers within which the parameter value is believed to fall (e.g: between 20% and 30% of the Dutch people are above 1,80m). The margin of error gives the lower border and the upper border of the margin.

A good estimator of a parameter has two properties:

1. Unbiased
   A good estimator has a sampling distribution that is centred at the parameter. A mean from a random sample should fall around the population parameter and this is especially the case with multiple samples and thus a sampling distribution.
2. Small standard deviation
   A good estimator has a small standard deviation compared to other estimators. The sample mean is preferred over the sample median, even in a normal distribution, because the sample mean has a smaller standard deviation.

An interval estimate is designed to contain the parameter with some chosen probability, such as 0.95. Confidence intervals are interval estimates that contain the parameter with a certain degree of confidence. A confidence interval is an interval containing the most believable values for a parameter. The probability that this method produces an interval that contains the parameter is called the confidence level. A sampling distribution of a sample proportion gives the possible values for the sample proportion and their probabilities and is a normal distribution if np is larger than 15 and n(1-p) is larger than 15. The margin of error measures how accurate the point estimate is likely to be in estimating a parameter.

CONSTRUCTING A CONFIDENCE INTERVAL TO ESTIMATE A POPULATION PROPORTION
The point estimate of the population proportion is the sample proportion. The standard error is the estimated standard deviation of a sampling distribution. The formula for the standard error is:

The greater the confidence level, the greater the interval. The margin of error decreases with bigger samples, because the standard error decreases with bigger samples. The larger the sample, the narrower the interval. If using a 95% confidence interval over time, then 95% of the intervals would give correct results, containing the population proportion.

CONSTRUCTING A CONDIFENCE INTERVAL TO ESTIMATE A POPULATION MEAN
The standard error for the population mean has the following formula:

The t-score is like a z-score, but a bit larger, and comes from a bell-shaped distribution that has slightly thicker tails than a normal distribution. The distribution that uses the t-score and the standard error, rather than the z-score and the standard deviation is called the t-distribution. The standard deviation of the t-distribution is a bit larger than 1, with the precise value depending on what is called the degrees of freedom. The t-score has

CATEGORICAL RESPONSE: COMPARING TWO PROPORTIONS:
Bivariate methods is the general category of statistical methods used when we have two variables. The outcome variable on which comparisons are made is called the response variable. The binary variable that specifies the groups is the explanatory variable. In an independent sample, observations in one sample are independent from observations in another sample. If two samples have the same subjects, they are dependent. If each subject in one sample is matched with a subject in another sample there are matched pairs and the data is dependent as well.

The formula for the standard error for comparing two proportions is:

A 95% confidence interval for the difference between two population proportions has the following formula:

The proportion (p̂) is called a pooled estimate, since it pools the total number of successes and total number of observations from two samples. This uses the presumption p1=p2. The test statistic uses the following formula:

The standard error for the test statistic uses the following formula:

QUANTITATIVE RESPONSE: COMPARING TWO MEANS:
The standard error for comparing two means has the following formula:

A 95% confidence interval for the difference between two population means has the following formula:

The confidence interval for the difference between two population means uses the t-distribution and not the z-distribution. Interpreting a confidence interval for the difference of means uses the following criteria:

1. Check whether or not 0 falls in the interval
   If it does, it could be that mean 1 is mean 2.
2. Positive confidence interval suggests that mean 1 – mean 2 is positive
   If the confidence interval only contains positive numbers, this suggests that mean 1 – mean 2 is positive. This suggests that mean 1 is larger than mean 2.
3. Negative confidence interval suggests that mean 1 – mean 2 is negative
   If the confidence interval only contains negative numbers, this suggests that mean 1 -  mean 2 is negative. This suggests that mean 1 is smaller than mean 2.
4. Group order is arbitrary
   It is arbitrary whether one group is group one or the other.

The test statistic of a significance test comparing two population means uses the following formula:

It uses minus zero because the null hypothesis is that there is no difference between the groups and is thus zero.

OTHER WAYS OF COMPARING MEANS AND COMPARING PROPORTIONS
If it is reasonable to expect that the variability as

MODEL HOW TWO VARIABLES ARE RELATED
A regression line is a straight line that predicts the value of a response variable ‘y’ from the value of an explanatory variable ‘x’. The correlation is a summary measure of association. The regression line uses the following formula:

The data is plotted before a regression line is made, because it can be strongly influenced by outliers. The regression equation is often called a prediction equation. The difference between y - ŷ, between an observed outcome y and its predicted value ŷ is the prediction error, called the residual. The average of the residuals is zero. The regression line has a smaller sum of squared residuals than any other line. It is called the least squares line. The population regression equation has the following formula:

This formula is a model. A model is a simple approximation for how variables relate in a population. The probability distributions of y values at a fixed value of x is a conditional distribution (e.g: the means of annual income for people with 12 years of education).

DESCRIBE STRENGTH OF ASSOCIATION
Correlation does not differentiate between response and explanatory variables. The formula for the slope uses the correlation and can be calculated as following:

Using this formula, the y-intercept can be calculated:

The slope can’t be used to determine the strength of the association, because it determines on the units of measurement. The correlation is the standardized version of the slope. The formula for the correlation is the following:

A property of the correlation is that at any particular x value, the predicated value of y is relatively closer to its mean than x is to its mean. If a particular ‘x’ value falls 2.0 standard deviations from the mean with a correlation of 0.80, then the predicted ‘y’ is ‘r’ times that many standard deviations from its mean, so the predicted ‘y’ would be 0.80 times 2.0 standard deviations from the mean. The predicted ‘y’ is relatively closer to its mean than ‘x’ is to its mean. This is regression toward the mean. If the first observation is extreme, the second observation will be more toward the mean and will be less extreme.

Predicting ‘y’ using ‘x’ with the regression equation is called the residual sum of squares and this uses the following formula:

The measure r squared is interpreted as proportional reduction in error (e.g: if r squared = 0.40, the error using y-hat to predict y is 40% smaller than the error using y-bar to predict y). The formula for r squared

COMPARE TWO GROUPS BY RANKING
Nonparametric statistical methods are inferential methods that do not assume a particular form of distribution (e.g: the assumption of a normal distribution) for the population distribution. The Wilcoxon test is the best known nonparametric method. Nonparametric methods are useful when the data are ranked and when the assumption of normality is inappropriate.

The Wilcoxon test sets up a distribution using the probability of each difference of the mean rank. This test has five steps:

1. Assumptions
   Independent random samples from two groups.
2. Hypotheses
   
   
   
3. Test statistic
   This is the difference between the sample mean ranks for the two groups.
4. P-value
   This is a one-tail or two-tail probability, depending on the alternative hypothesis.
5. Conclusion
   The null hypothesis is either rejected in favour of the alternative hypothesis or not.

The sum of the ranks can also be used, instead of the mean of the ranks. When conducting the Wilcoxon test, a z-test can also be conducted if the sample is large enough. This z-test has the following formula:

A Wilcoxon test can also be conducted by converting quantitative observations to ranks. The Wilcoxon test is not affected by outliers (e.g: an extreme outlier gets the lowest/highest rank, no matter if it’s a bit higher or lower than the number before that). The difference between the population medians can also be used if the distribution is highly skewed, but this requires the extra assumption that the population distribution of the two groups have the same shape. The point estimate of the difference between two medians equals the median of the differences between the two groups. A sample proportion can also be used, by checking what the proportion is of observations in group one that’s better than group two. If there is a proportion of 0.50, then there is no effect. The closer the proportion gets to 0 or 1, the greater the difference between the two groups.

NONPARAMETRIC METHODS FOR SEVERAL GROUPS AND FOR MATCHED PAIRS
The test for comparing mean ranks of more than two groups is called the Kruskal-Wallis test. This test has five steps:

1. Assumptions
   Independent random samples.
2. Hypotheses
   
   
3. Test statistic
   The test statistic is based on the between-groups variability in the sample mean ranks. The test statistic uses the following formula:
   The test statistic has an approximate chi-squared distribution with g-1 degrees of freedom.
4. P-value
   The right-tail probability above observed test statistic value from chi-squared distribution.
5. Conclusion
   The null hypothesis is either rejected in favour of the alternative hypothesis or not.

It is