Your study guide

0.1What statistics is for

Suppose you want to know how much weight people lose on a diet. You cannot test every person on Earth, so you test a few and try to say something about all of them. That jump, from a few to all, is what statistics is about.

The next table is worth learning now, because the lectures use these letters on almost every page. Greek letters are the true values. Latin letters are what you calculate.

A hat always means "an estimate of". So $\hat{\mu }$ is an estimate of $\mu .$

Statistics is never about perfect data. If you have perfect data, you need the probability, not the statistics. Statistics is about imperfect data: how to gain information, not perfect but approximate information, from approximate data.

Lecturer L1 38:23

0.2How to read the symbols

Formulas look hard mostly because of the notation. Each symbol is a short way of saying something simple.

0.3The centre of the data: mean and median

The first question about any data is: what is a typical value? We use a real sample from Lecture 3: ten people had to guess when one minute had passed, in seconds.

Why have two measures of the centre? Suppose somebody mistypes 66 as 166.

0.4The spread of the data: variance and standard deviation

A centre is not enough. Two classes can have the same mean mark, but in one class everybody is close to the mean and in the other class the marks are all over the place. We need a number for how far values usually are from the mean.

Build it step by step on a tiny sample, $x=(2,4,4,5,10),$ with mean 5:

1. Find each deviation from the mean.
2. Square them. The plain deviations always add up to 0, so their average tells you nothing. Squares are never negative.
3. Add the squares. This is the sum of squares, here 36. ANOVA is built on this quantity.
4. Divide by $n-1.$ This gives the variance: $s^2=\frac{36}{4}=9.$
5. Take the square root. This gives the standard deviation: $s=\sqrt{9}=3.$

Why the square root? Squaring also squares the unit. If the data are in kilograms, the variance is in kilograms squared, which means nothing to a person. The standard deviation is back in kilograms. So we calculate with the variance, and we talk about the standard deviation.

Why $n-1$ and not $n?$ See Good and bad guesses below.

x <- c(53, 48, 45, 55, 63, 51, 66, 56, 50, 58)
mean(x)
[1] 54.5
var(x)
[1] 42.94444
sd(x)
[1] 6.553201

If you say 3 kilograms, it's almost useless information in statistics. If you add the information, it is 3 kilograms with standard deviation of 100 grams, this is a proper statistical result.

Lecturer L1 41:05

0.5Quartiles, box plots and histograms

Quartiles cut the sorted data into four equal parts. Q1 has about 25 % of the values below it, the median (Q2) has 50 %, and Q3 has 75 %. The interquartile range, IQR = Q3 − Q1, is the width of the middle half of the data.

• The whiskers reach to the most extreme values that are still at most 1.5 × IQR from the box.
• Points beyond the whiskers are drawn one by one, and they are possible outliers.
• A box plot shows the median, not the mean.

quantile(x)
   0%   25%   50%   75%  100%
45.00 50.25 54.00 57.50 66.00
IQR(x)
[1] 7.25

A histogram shows the shape of the data: cut the axis into bins, and draw a bar for how many values fall in each one. The shape tells you how the mean and median relate:

0.6Probability from zero

A probability pravděpodobnost is a number from 0 to 1. It says how often something would happen if you repeated the situation many times. $P=0.5$ means "about half of the time". Everything in statistics is built on four rules.

Independent means that knowing one result tells you nothing about the other. Two rolls of a die are independent. Your height and your weight are not.

0.7Random variables, expected value and variance

A random variable is a number whose value depends on chance, like the result of a roll. Its distribution lists the possible values and how likely each one is.

The expected value střední hodnota $E(X)$ is the long-run average: each value times its probability, all added up. For a die:

You can never roll 3.5. It is the average of many rolls. The variance is the expected squared distance from the mean:

These are the same ideas as $\bar{x}$ and $s^2,$ but for the whole theoretical distribution. So $E(X)$ is $\mu ,$ and $\mathrm{Var}(X)$ is ${\sigma }^2.$

The binomial distribution counts successes in $n$ independent tries, each with success probability $p.$ We write $X\sim \mathrm{Bin}(n,p).$ Then $E(X)=np$ and $\mathrm{Var}(X)=np(1-p).$ For 10 fair coin flips, $E(X)=5$ and $\mathrm{Var}(X)=\mathrm{2.5.}$ The sign test in Lecture 3 and the chi-squared test in Lecture 4 both rely on this.

0.8Continuous variables: probability is an area

Height can be any number, like 170.2371 cm. The chance of exactly that value is 0, so bars for single values make no sense. Instead we ask about ranges, like "between 160 and 180 cm".

Imagine a histogram of many people's heights. With more data and narrower bins, the top of the bars turns into a smooth curve, called the probability density function (PDF) hustota pravděpodobnosti.

If you said that both areas are equal to one, I hope everybody can see that the red area is much bigger than the yellow area. But you must look at the scale. It is mandatory: each mathematical plot must have the scales on the axis.

Lecturer L1 86:06

0.9The normal distribution and z-scores

The normal distribution normální rozdělení, also called Gaussian, is the famous bell curve. Two numbers describe it fully: the mean $\mu$ says where the centre is, and the standard deviation $\sigma$ says how wide it is. We write $X\sim N(\mu ,{\sigma }^2).$

To compare normal variables, we convert them to a common scale. The z-score standardizovaná hodnota says how many standard deviations a value is from the mean:

$N(0,1)$ is the standard normal distribution. So a question about any normal variable becomes a question about a single curve.

1 - pnorm(185, mean = 170, sd = 10)
[1] 0.0668072

qnorm(0.90, mean = 170, sd = 10)
[1] 182.8155

R uses the same four letters for every distribution. Learn these, and you know the functions for all of them. Replace norm with t, chisq, f, binom or exp.

Where the number 1.96 comes from

You will see 1.96 all the time. It cuts off the middle 95 % of the standard normal curve, which leaves 2.5 % in each tail.

Read this subscript carefully. For a middle area of $1-\alpha ,$ the upper limit has $1-\frac{\alpha }{2}$ on its left. With $\alpha =0.05,$ that is $\mathrm{0.975.}$ This is why you see $t_{1-\alpha /2}$ in nearly every formula of the course.

0.10The key idea: the sample mean is random too

This is the most important idea of the whole course, and it is the one most people miss. Test 10 people and you get a mean of 54.5 seconds. Test 10 other people and you get a different mean. So the sample mean is itself a random variable, and it has its own distribution. That distribution is called the sampling distribution výběrové rozdělení.

How wide is it? Use the rules from Random variables. Each $X_i$ has mean $\mu$ and variance ${\sigma }^2,$ and they are independent.

0.11The central limit theorem

The last topic told us the centre and the width of the sampling distribution. What about its shape? This is the surprising part.

Try it. Start with "Long tail" and $n=1,$ which is nothing like a bell. Now raise $n$ to 5, 30 and 100.

Once you have independent observations and you take average of the independent observations, after standardization, it means take the value minus the mean and divide it by the standard deviation, you will obtain something close to standard normal distribution. The central limit theorem is saying that regardless of the probability distribution of the original variable, after taking the average of many, many values, you are going more and more close to the normal distribution.

Lecturer L3 48:47

This theorem is everywhere in the course:

• ANOVA and t-tests are robust to non-normal data, because they really work with means (Lecture 2).
• The binomial looks normal for large $n$, because a count is a sum of zeros and ones. This special case is the de Moivre-Laplace theorem used in the sign test and the chi-squared test.
• Asymptotic versions of the rank tests in Lecture 3 use it too.

0.12Good and bad guesses: estimators

An estimator odhad is a rule for guessing a parameter, like "take the sample mean". There are two kinds of estimate:

• A point estimate is one number: "the mean is 54.5".
• An interval estimate is a range: "the mean is between 49.8 and 59.2". This is a confidence interval.

What makes an estimator good?

Unbiased estimator is an estimator whose expected value is exactly equal to the theoretical value. Imagine a case that you have 1000 samples from the same population. If you really use 1 over n minus 1 and you evaluate the sample variance on each sample, and take an average of these 1000 values, this will be a good estimator of the variance from the theory.

Lecturer L5 9:01

0.13Confidence intervals: the idea

You measured 10 people and got a mean of 54.5 seconds. Is the true mean exactly 54.5? Almost certainly not, because another sample would give another number. A single number hides this uncertainty. A confidence interval (CI) interval spolehlivosti shows it, by giving a range of believable values for the parameter.

0.14Building a 95 % interval, step by step

Suppose for a moment that we know $\sigma .$ From the last topics, we know how the sample mean behaves:

1. The sample mean is normal around the true mean, with standard deviation $\sigma /\sqrt{n}.$ So after standardizing: $$Z=\frac{\bar{X}-\mu }{\sigma /\sqrt{n}}\sim N(0,1).$$
2. 95 % of the standard normal lies between ±1.96: $$P(-1.96\le \frac{\bar{X}-\mu }{\sigma /\sqrt{n}}\le 1.96)=\mathrm{0.95.}$$
3. Multiply everything by $\sigma /\sqrt{n}:$ $$P(-1.96\frac{\sigma }{\sqrt{n}}\le \bar{X}-\mu \le 1.96\frac{\sigma }{\sqrt{n}})=\mathrm{0.95.}$$
4. Rearrange to get $\mu$ alone in the middle: $$P(\bar{X}-1.96\frac{\sigma }{\sqrt{n}}\le \mu \le \bar{X}+1.96\frac{\sigma }{\sqrt{n}})=\mathrm{0.95.}$$

The picture shows why this works. The sample mean lands within 1.96 SE of $\mu$ in 95 % of samples. Exactly in those samples, an interval of ±1.96 SE around $\bar{x}$ reaches $\mu .$

0.15The real case: σ is unknown, so we use t

In practice nobody tells you $\sigma .$ You estimate it with $s.$ But $s$ also changes from sample to sample, which adds extra uncertainty. William Gosset, working at the Guinness brewery and writing as "Student", worked out the exact result in 1908:

The Student t distribution looks like the normal curve, but with heavier tails. It depends on the degrees of freedom, here $n-1.$ With more data, $s$ becomes reliable, and $t$ approaches the normal curve.

qt(0.975, df = 9)
[1] 2.262157
t.test(x)$conf.int
[1] 49.81212 59.18788
attr(,"conf.level")
[1] 0.95

0.16What "95 % confident" really means

This is the most common mistake in all of statistics, and a favourite exam question. The true mean $\mu$ is a fixed number. It is not random. Before you collect data, the interval is random: it depends on the sample that you will get.

Press "Draw 30 new samples" a few times. Each time, the red lines are different, but about 5 % of the lines miss. In real life you get only one line, and you can never know whether yours is one of the red ones.

0.17What makes an interval narrow or wide

A narrow interval tells you more. Look at the margin of error, $t_{1-\alpha /2}\cdot \frac{s}{\sqrt{n}}.$ Three things control it.

The 99 % interval contains 60, and the 95 % interval does not. This is why the level must be fixed before you look at the data.

0.18Comparing two groups: rebuilding R's interval

Very often the parameter of interest is a difference, like ${\mu }_A-{\mu }_C.$ The form is the same: estimate ± critical value × standard error. Only the standard error changes. For independent samples, variances add (the rule above):

Paired data

If the same people are measured twice, before and after, the two samples are not independent. Calculate the difference for each person, and use the one-sample interval on those differences. The formula above would be wrong, because it adds variances as if the samples were independent.

One-sided intervals

A one-sided test has a one-sided interval, open on one side. All of $\alpha$ goes to one tail, so the critical value uses $1-\alpha ,$ not $1-\frac{\alpha }{2}.$

A trap: overlapping intervals

0.19Confidence intervals for other parameters

Every interval in the course follows the same recipe. Once you know the recipe, you only have to look up three things: the estimate, its standard error, and which distribution gives the critical value.

0.20The idea of a statistical test

A test is like a court trial. The accused is presumed innocent until the evidence is strong enough. We start by assuming "nothing special is going on", and ask whether the data make that assumption hard to believe.

binom.test(9, 10, p = 0.5)$p.value
[1] 0.02148438

That is the logic of every test in this course. Only the statistic changes. With 8 heads you would get $p=0.109,$ so you would not reject. This is the same calculation as in the sign test.

0.21The p-value and your first t-test

Most test statistics have the same form. They ask: how far is the estimate from what $H_0$ claims, measured in standard errors?

A value near 0 agrees with $H_0.$ A large value, positive or negative, speaks against $H_0.$

Statistically significant does not mean important

"Significant" only means "unlikely to be pure chance". Suppose a diet changes weight by 0.2 kg, with $\sigma =5$ kg and 10,000 people in each group. Then $T=\frac{0.2}{5\sqrt{2/10000}}=2.83$ and $p=\mathrm{0.005.}$ This is highly significant, but 0.2 kg is useless in practice. So report the effect size too, and a confidence interval is the best way to show it.

Errors and power

A test can be wrong in two ways. Type I: reject a true $H_0,$ a false alarm, with probability $\alpha .$ Type II: fail to reject a false $H_0,$ a missed effect, with probability $\beta .$ Power $=1-\beta$ is the chance of finding an effect that really exists. More data, a larger effect and less noise all raise the power. Lecture 1 covers this in detail.

0.22Reading R's t-test output, line by line

t.test(x, mu = 60)

        One Sample t-test

data:  x
t = -2.6541, df = 9, p-value = 0.0263
alternative hypothesis: true mean is not equal to 60
95 percent confidence interval:
 49.81212 59.18788
sample estimates:
mean of x
     54.5

0.23The distributions you will meet

You do not need to memorize their formulas. Know what each one is for, what it looks like, and where it comes from.

The first four are one family, built from the normal one step at a time. Lecture 1 builds each of them from the same small sample.

0.24Are you ready for Lecture 1?

Try to answer each question out loud before you open the answer. If one is hard, go back to its topic. Lecture 1 will then feel like revision.

1.1Probability, statistics and machine learning

These three words are often mixed. Here is the difference.

Czech names: probability theory teorie pravděpodobnosti, mathematical statistics matematická statistika, machine / statistical learning strojové / statistické učení.

Statistics is never about perfect data. If you have perfect data, you need the probability, not the statistics. Statistics is about imperfect data: how to gain information, not perfect but approximate information, from approximate data.

Lecturer L1 38:23

If you say 3 kilograms, it's almost useless information in statistics. If you add the information, it is 3 kilograms with standard deviation of 100 grams, this is a proper statistical result.

Lecturer L1 41:05

1.2The three kinds of statistical work

Doing statistical inference without looking at the data is a bad practice. Every time you load new data, before you do anything else, you should do these two things very quickly: make some tables, make some typical visualizations. You should know your data.

Lecturer L1 43:50

1.3Parametric, nonparametric, semiparametric

Statistical inference has three families of methods.

The parametric methods are relatively easy to calculate and to interpret. Everybody likes them, but they have relatively strict assumptions. And in reality, unfortunately, many people are using methods which should not be used. You should be sure that you understand the assumptions of each method.

Lecturer L1 45:44

1.4Statistical data types

Stevens (1946) sorted data into four levels. Each level allows everything from the level before it, plus something new.

Two special kinds of nominal data:

• Dichotomous data: only 2 categories, often FALSE and TRUE.
• Dummy variable: the numbers 1 and 0 show membership in a category. You need it later for regression, because matrices can only hold numbers.

The temperature expressed in degrees of Celsius does not have a fixed zero. If you say that today is two times higher temperature than yesterday, if it means that today it is 20 degrees Celsius and yesterday it was 10 degrees Celsius, this is not correct. If you say the same in Kelvins, you are correct.

Lecturer L1 53:19

Another typical example is the calendar year. Do you remember that there is no year zero?

Lecturer L1 54:16

Two more notes: you need to know the meaning of the data (the variables, the experiment, the preprocessing) to choose the type. And continuous data can be turned into ordered categories by sorting them into intervals, with some loss of information.

If you miss this, your statistical analysis will be completely incorrect, because the proper assignment of the statistical data types is saying to the functions what they should do with the data.

Lecturer L1 69:07

Discrete or continuous

This second classification asks if you can count the possible values.

• Discrete: the values can be listed: finitely or countably many, often integers. Examples: gender, blood type, grades, ranking, number of children.
• Continuous: uncountably many values, typically a whole interval, and arbitrarily precise. This applies to interval and ratio data. Examples: height, weight, time.

1.5Tidy data and the tidyverse

A table is tidy data (Hadley Wickham, 2014) when:

• every column is a variable, with one unique meaning;
• every row is an observation (one case);
• every cell is a single value. (Newer "nested data" allows a column that holds lists or smaller tables.)

The goal: "Spend less time fighting with the tools and more time working on your analysis." In R a table is a data.frame (old and widely compatible) or a tibble, also written tbl_df (the modern version). All tidyverse functions expect tidy tables.

Everything should be a table. To be really correct, even if you have only one observation, it should be a table with one row and one column.

Lecturer L1 58:05

The tidyverse (2016) is a collection of libraries for data science that share one philosophy, one consistent syntax, and work well together. The pipe |> or %>% makes code read like a sentence.

In classical, modern statistical data analysis, you should completely avoid writing loops, because everything can be nowadays done without loops.

Lecturer L1 70:27

dplyr and tidyr verbs

These work almost like SQL.

ggplot2

You declare what you want in four steps: give tidy data, map columns to aesthetics with aes (x-axis, y-axis, grouping, line type, color, background, symbol), say which shapes to draw, then adjust details. Start with ggplot and add layers with +.

• Simple shapes: geom_line, geom_point, geom_jitter.
• Statistical shapes: geom_boxplot, geom_histogram, geom_density, geom_bar, geom_ribbon, geom_qq, geom_qq_line, geom_rug, geom_area.

This will be the minimum accepted graphics for you. We will really not accept any graphics in basic R.

Lecturer L1 63:03

1.6The running example: weight loss on three diets

78 people followed diet A, B or C for 6 weeks. The lectures keep coming back to these data.

str(data)   # or: glimpse(data)
'data.frame': 78 obs. of 7 variables:
 $ Person      : int  25 26 1 2 3 4 5 6 7 8 ...
 $ Gender      : int  NA NA 0 0 0 0 0 0 0 0 ...
 $ Age         : int  41 32 22 46 55 33 50 50 37 28 ...
 $ Height      : int  171 174 159 192 170 171 170 201 174 176 ...
 $ WeightPrev  : int  60 103 58 60 64 64 65 66 67 69 ...
 $ Diet        : int  2 2 1 1 1 1 1 1 1 1 ...
 $ Weight6Weeks: num  60 103 54.2 54 63.3 61.1 62.2 64 65 60.5 ...

# Add the weight loss, and give Diet and Gender their proper statistical type
data <- data |>
  mutate(
    WeightLoss = WeightPrev - Weight6Weeks,
    Diet = factor(Diet, labels = c("A", "B", "C")),
    Gender = factor(Gender, labels = c("Female", "Male"))
  )

This is a local work inside a tidy table. There's an expression on the right hand side, and the result is stored into a column with the name on the left hand side. If the column does not exist, it is automatically created. If it already exists, it is rewritten.

Lecturer L1 68:00

Descriptive statistics by group

data |>
  group_by(Diet) |>
  summarise(size = n(), mean = mean(WeightLoss), st.dev = sd(WeightLoss),
            variance = var(WeightLoss), median = median(WeightLoss))

Diet C looks better: about 2 kg more. Is that a real effect or just luck? Tests answer this question.

Everybody should understand or remember what is a box plot. [The middle line is the] median. [The box edges are the] so-called quartiles. How many of the observations are inside the box? 50% of the data.

Lecturer L1 73:54

1.7Random variable and probability distribution

A random variable náhodná veličina $X$ is a measurable mapping $X:\Omega \to \mathbb{R}$ from the space $\Omega$ of elementary events to the real numbers. A realization (or observed value) $x$ is the value that really happened in the experiment.

It's a connection between the space of outcomes of some experiment in the real world and the world of numbers. To be able to use any part of mathematics, we must transform everything into numbers.

Lecturer L1 76:42

A probability distribution rozdělení pravděpodobnosti $P$ assigns a probability to each event (each set of values). It describes the type of randomness of the phenomenon.

Distributions come in two groups:

1.8Random sample and independence

A random sample náhodný výběr of size $n$ (also called i.i.d., independent and identically distributed) is a random vector $(X_1,\dots ,X_n)$ where the variables

• have the same probability distribution $P,$ and
• are mutually independent.

This is a cornerstone of classic statistics. Classic methods assume that your data form a random sample.

It is not the independence that we have done something independently, as we are typically saying. [It] has a precise definition given by the probability theory, and the definition is very strict.

Lecturer L1 80:08

Therefore it is very difficult to be tested. In fact, we have no general statistical method to test independence, because it is about the origins of the whole theory.

Lecturer L1 81:24

1.9Normal distribution and standardization

From here on, we build the whole family of distributions from one simulated example: 5 values $Y_1,\dots ,Y_5$ from $N(\mu =3,{\sigma }^2=2^2).$

Every time you will hear about standardization, it always means: take your data, separately each observation, minus mean, divided by standard deviation.

Lecturer L1 83:43

Common questions, answered

• What is the probability density function (PDF)? A function $f(x)\ge 0$ whose total area is 1: ${\int }_{-\infty }^{\infty }f(x)dx=1.$
• Where is the probability on the density plot? Probability is an area under the curve, not the height of the curve.
• What is a histogram? A picture of real data: split the axis into bins and draw a bar for each one. With the density scale, the bars have total area 1, so it estimates the PDF.
• What is the 95 % quantile $q_{0.95}?$ The number with 95 % of the probability on its left: $P(X\le q_{0.95})=\mathrm{0.95.}$
• How do you find it on the plot? Find the place on the x-axis where the area to the left is 0.95.

If you said that both areas are equal to one, I hope everybody can see that the red area is much bigger than the yellow area. But you must look at the scale. It is mandatory: each mathematical plot must have the scales on the axis.

Lecturer L1 86:06

1.10Chi-squared distribution and degrees of freedom

Step 1. Standardize each value, square it, and add everything up.

Step 2. In practice we do not know $\mu .$ Replace it with its unbiased estimator, the sample mean $\bar{Y}=\frac{1}{n}{\sum }_{i=1}^n{Y}_i.$

Why $n-1?$

The sample mean is itself a random variable, because it comes from the random sample. It also creates one constraint:

Because we are using the data twice: once inside this square, and second time to calculate this sum. You are using the same information in different ways. And in statistics, you cannot do this without changing the degrees of freedom. Every time you use the estimated mean, the sample mean, the average, you are losing one degree of freedom.

Lecturer L1 90:40

In statistics, most of the work is done via the quantiles. Maybe the shapes are similar. It depends on the point of view. But these two numbers are very different. You cannot say that instead of this quantile, you will use this quantile. If you do this, you are saying completely incorrect results.

Lecturer L1 93:41

1.11From the normal distribution to Student's t

Step 1. Now only center the values, and add them:

Why? A sum of independent normal variables is normal. Means add: $0+\cdots +0=0.$ Variances add: ${\sigma }^2+\cdots +{\sigma }^2=n{\sigma }^2.$

Step 2. Standardize $D:$ divide by its standard deviation $\sigma \sqrt{n}.$

This is the formula of the Z test: it tests a mean when you know $\sigma .$

Step 3. We rarely know $\sigma .$ Use the sample standard deviation:

$S_Y$ is a random variable, because it depends on the sample. So the result is no longer normal:

There is no magic. It's only some work. Put everything from the previous [steps] together until you obtain this final, so-called t statistics. This is exactly how the test is working. In fact, this is almost the whole mathematical proof.

Lecturer L1 97:16

You should at least remember that you know normal distribution, standardized distribution, chi-square distribution, Student t distribution and Fisher distribution.

Lecturer L1 98:07

1.12Cumulative distribution function

The CDF gives the probability of being at most $x:$ $F(x)=P(X\le x).$ It goes from 0 to 1 and never decreases.

• From PDF to CDF: $F(x)$ is the area under the density to the left of $x.$ So $F(x)={\int }_{-\infty }^{x}f(t)dt.$
• From CDF to quantile: the quantile reverses the CDF. Pick a probability on the y-axis, read $x$ on the x-axis. $q_p=F^{-1}(p).$

Plotting the CDFs of all six variables together shows an important message: you must understand the correct type of distribution of a random variable, because each one has a different curve and different quantiles.

There is one important note: they are using a different mathematical definition of the CDF, in Czech the distribution function. Be very careful.

Lecturer L1 34:31

1.13Is my data normal? Histogram and Q-Q plot

Back to the diet data.

data |> ggplot(aes(x = WeightLoss, fill = Diet, color = Diet)) +
  geom_histogram(aes(y = after_stat(density)), binwidth = 2, linewidth = 0.3,
                 color = "black", alpha = 0.7) +
  labs(x = "Weight Loss / kg", y = "density") +
  facet_grid(Diet ~ .)

data |> ggplot(aes(sample = WeightLoss, color = Diet)) +
  geom_qq(distribution = qnorm, size = 2.5) +
  geom_qq_line(distribution = qnorm, linewidth = 0.7, color = "black") +
  labs(x = "theoretical quantiles of N distribution", y = "sample quantiles") +
  facet_grid(. ~ Diet)

1.14Hypotheses and decisions

The history: Pearson, Gosset and Fisher started significance testing around 1900. Neyman and Pearson (1933) created the modern theory.

A statistical hypothesis is a pair of statements:

• $H_0,$ the null hypothesis nulová hypotéza: the statement being tested. Often "no effect" or "no difference", but not only.
• $H_1,$ the alternative hypothesis alternativní hypotéza: the statement tested against $H_0.$ Usually "there is an effect".

A statistical test is a decision based on a random sample. There are exactly two results:

• the test rejects zamítá $H_0$ in favor of $H_1,$ or
• the test does not reject (fails to reject) nezamítá $H_0$ in favor of $H_1.$

The general theory is the Neyman-Pearson lemma. A good criterion for choosing hypotheses is a likelihood ratio.